Can an electron go along different paths at once in a circuit A single electron moves along a circuit and comes to a fork in the wires.  The wires separate but come back together near the end of the circuit.  From what I know, the electron will travel along all the possible paths there are due to its wave property.  So does this mean that the electron is physically traveling both paths at once?
 A: The electron crawls along very slowly. And long before it gets to the diodes other electrons flow through the diodes which then both emit light.
Why? Because the electrons in front of it feels the field from the electron before the electron gets to it and they move forwards and when they move the electrons in front of them move, etc. So some other electrons get to the fork long before the first electrons does and those push electrons along both paths of the fork.
Electricity and current flows from one end of a circuit to another much faster than the electrons move around from one end of the circuit to another. 
An analogy would be a super crowded room you might jostle your neighbors on accident as you move towards the door. And that might make them jostle their neighbors and you might even see people by he doors get jostled from the indirect effects and that could happen long before you get to the door yourself. News and fields can travel faster than matter, they can travel at lightspeed and matter is more limited in how it moves.
Response to new question that invalidates my answer to the original question
The short answer is yes and no. Yes in some sense it does. And no in the sense that you can't detect it and the whole question really is based on assuming a misconception that an electron follows a path through some fixed space while there is a bunch of stuff in that fixed space. And that misconception hides the reason you can't detect it.
When people first introduce quantum mechanics they talk about just one particle and then people develop completely wrong ideas about absolutely everything and sometimes some of the wrong ideas never get fixed for that person. This is unfortunate.
Important fact 1
All wavefunctions are for multiple particles and they assign complex numbers to configurations of all the particles not to places in space. What is a configuration? Imagine a point in the xy plane and we say the x coordinate tells us where one particle is and the y coordinate tells us where the second particle is. So for instance the line x=y corresponds to the configurations where the particles are on top of each other and he half plane where y>x corresponds to the configurations where the second particle is to the right of the first particle. Learn to see a point in xy plane as specify two particles. Noe if you have n particles you have to specify a point in a 3n dimensional space to say where they all are and vice versa. The wavefunction is not defined in real 3d space like a temperature field or a pressure or density field or like a classical electromagnetic wave. It is assigns complex numbers to whole configurations.
So there are many configurations that correspond to one electron in one wire and these different configurations are different and pretending they are the same will lead to incorrect results. So we have to watch out.
Important fact 2 
Interference only happens when a wave traveling through configuration space and another wave travelling through configuration space travel towards the same region of configuration space and actually overlap in configuration space.  If just one particle in the universe is in the different location the configurations are different they are different points in the 3n dimensional space and the waves are not overlapping and no interference happens. So now let's see what is going on.

A single electron moves along a circuit and comes to a fork in the wires. 

So it isn't just a single electron, its an electron and its also all the protons and a neutrons and gluons and (other) electrons in the wire. So we actually have a huge n and so the configuration space is in a huge 3n dimensional space.

The wires separate but come back together near the end of the circuit. 

If the electron is going to stay in the wires rather than just heading the direction it was going then it is interacting with the things in the wire. Which means the things in the wire are also interacting with the electron. This is important.

From what I know, the electron will travel along all the possible paths there are due to its wave property. 

Not right. There is a wave but it is a wave of configurations of all the particles. It is an assignment of a complex number to every configuration of all the particles, not just the electron. So the path isn't a path of the electron through real space the path is a path in configuration space so it is entirely possible that the path evolves so the thibgs in the wire change too. And in this situation this will happen.

So does this mean that the electron is physically traveling both paths at once?

Yes and no. Let's see what happens. Assume at some point the two branches of the wire are parallel and going in the x direction. And pick a section of wire say that has xyz coordinates of (1,0,0) and another that has coordinates of (1,2,0) then we can pick one of the particles on each part of the wire.
Lets measure everything in meters. This means that in the configuration of all the particles you have something like $(0.99,0,0,1,0,0,1,20, ...)$ where the first three are saying the electron is at (0.99,0,0) and there is a particle at (1,0,0) and they are close and there is a third particle at (1,2,0) which is in a completely different section of wire. And the dots mean we aren't going to bother mentioning where all the other particles are. Since the wire goes in the x direction that 0.99 could become 0.999 and get even closer to the particle that was already in the place (1,0,0). Eventually it can get close enough that they can interact strongly. And when that happens the configuration of both can change.
So let's visualize this by only looking at the x coordinate of each particle.  You can imagine a point in 3d space and the xyz of (0.99,1,1) and the x=0.99 can tell you the x of the moving electron is 0.99 and the y=1 could tell you that the particle near the electron has an x of 1 and the z=1 could tell you that the electron far away from the moving one has an x that is also 1.
The your point (0.99,1,1) could evolve to (0.999,1.01,1) it is pushing the particle in front of it forwards a bit. And this is different than the classical version I gave before. The point here is that it didn't change the far away particle. But if the moving electron went around and later ends up with a configuration like (20,1.01,1) where the moving electron is now in a section of wire where the two wires have connected back together.
And now we get to the point. The wave can and does split and can and does give nonzero complex numbers for configurations where it went left and configurations where it went right. However when the moving electrons that went left combine with the moving electrons that went right the nonzero complex numbers are assigned to final configurations like (20,1.01,1) where a part of the wire on one branch was jostled and configurations like (20,1,1.01) where a part of the wire on other branch was jostled and this means the two options don't interfere.
And it's actually worse. Those jostled pieces of the wire jostle their neighbors and so on. So  you can just have pieces on either side interact with each other in a way where they have the same configuration like (20,1.005,1.005) because one so many particles have been affected there is no way to naturally get them to forget.  And that's really it if you have changed things I'm your passing and the changes are different for the two options then it is like the universe remembers and they are different configurations and they won't interfere. So each configuration out there acts like the other one doesn't exist.
And you can't force them to forget. Because if you make them forget then noe you ever forcing them so the universe remembers in you. So it has to be natural not just you cleaning up the mess the electron made after it went through.
So it did go down both. But each acts like it only went down one. And that happens because the electron is making a mess as it goes down the wire and the different messes are distinguishable. If you sent an electron through a thin sheet that reflects half and lets half through and does so in a way where it isn't itself different when those two things happen (this is possible I'll explain that later) and then the two configurations of going through and being reflected could evolve to meet magnetic fields or gravitational fields or slits that bend them they could be directed to the same location. If that happens then the total configuration is the same and they can interfere.
So the real thing that is going on is there a change in the values of the complex numbers at each point. So it's not just the complex number at one configuration its really a whole region configurations like a box in that giant space of configuration. That box has nonzero complex numbers assigned to every point in the box (every configuration where no particle is too different) and that the spatial variation of the complex numbers (near some configuration) is balanced against the forces that configuration would produce and if they don't match right there then the complex number right there at that configuration changes.
So you can imagine a wave of nonzero values coming in towards a configuration of the electron hitting the thin half reflecting sheet. And then you get nonzero values for the configuration where the electron is hitting the thin sheet. And then you start getting two nonzero beams corresponding to 1) nonzero values for configurations where the electron bounces off and has 2) nonzero values for configurations where the electron passes through.
And since the box in configuration space where the values are nonzero is basically elongating in the direction corresponding to the position of the moving electron and then  splitting into two beams. The two beams together can interact with the thin sheet so that it is left in the same configuration.
If you want to try to visualize it not in configuration space you can imagine tracking each initial point in configuration space and then just seeing where it goes by imagining a dynamically changing configuration for everything exactly as if there were forces acting on particles.  
If you do that you see a configuration where there is like a tube of locations where the electron could be and if your trajectory is closer to the thin sheet than the average part of the tube you move as if there was a force from the phantom places were the electrons could have been and this force overcomes the actual classical repulsion of the thin sheet and it pushes you through but you did slow down as you passed through.  But if you follow a trajectory that started out farther than average from the thin sheet it is like that configuration feels a force from phantom locations that electrons closer to the sheet would have slowed down for some reason (as if some sheet were pushing on them to slow them down) and so you get pushed away from the sheet.  So you see trajectories where electrons bounce off before they even get to the sheet and the farther they started out from the sheet the farther away they bounced away. As if phantom electrons closer to the sheet had to push this electron out of their way to themselves be able to bounce off. And ones that start out closer get slowed down by the sheet but then act like there is a pile up of phantom electrons that get behind it as it slows down and as if that pile up pushes it harder until it bursts through the other side. And you see some in the middle that get really close to the thin sheet as if there were phantom electrons on both sides pushing it and it pushing them away and those configurations stay near the thin beam for a long long time (it depends on how long that initial beam of possible initial configurations was) but eventually when you stop having configurations that started out farther down the beam line those streamlines closest to the middle of the beam finally go through or reflect.
These are the streamlines of configurations that you see if you track the current through the configuration space. And it is right there in the Schrödinger equation (and it's continuity equation to track the stream lines). The thing I promised was to say that the beam could split without altering the thin sheet.
The beam started out with a box of initial configurations. And it had some thickness in every direction of the configuration space. So some of those different configurations correspond to different initial configurations of the thin sheet. When the thin sheet acts like it is exerting a force on the electron then it acts like it feels an equal and opposite force. So you'd naively think it's configuration would change. But first it acts like there are other forces two. Forces from phantom different initial configurations of the sheet getting piled up. Plus secondly even of changes configuration if two configurations get swapped it isn't a big deal. So maybe the configuration where the sheet started out arranged like A becomes a configuration where it is arranged like B. But maybe also the configuration where the sheet started out arranged like B becomes a configuration where it is arranged like A. This isn't a big deal. It's like current flowing in a perfect circle. The streamlines of the charges went in a circle but the charge density still didn't change.  So the configurations took up a region in configuration space where the wave was nonzero and they might move around but take up the same space. Like gas moving around in a box or condensed milk in a can of condensed milk. As long as the regions are nonzero then when two nonzero regions flow into the same region then they affect each other.
So the real reason the wires are a problem is because it isn't a thin sheet. In a thin sheet you have a chance to have a configuration where you just exert elastic type forces on it and don't permanent deform it and the splitting allows the two parts of beams to get momentum from each other rather than making the thin sheet recoil.
In the wire you have so many things to hit as you travel down the long way, you are going to deform something thing. And it is even worse you slow down if the wire has resistance and to keep up your current it has to get energy from the electromagnetic field, so it changes that too in addition to changing parts of the wire.
In the end this is what happens.
The electron changes things and does it in a way where such a huge part of the world is so different that the families of paths in configuration space that can be grouped into went-left paths and went-right paths lose the ability of the two groups to overlap in configuration space. Wow. Whole families of paths in a giant configuration space of all the particles. And quantum effects happen when the dynamics the different paths exert quantum forces based on how the different paths pile up and those forces are strong compared to the forces that configuration would feel. So they have to pile up to affect each other. So they must overlap.
And when I say path I mean the streamline of continuity equation of the Schrödinger equation because that is what we measure in the lab. If you want to do path integrals that is a mathematical sum of classical paths.
A: Here's the problem: scattering.  It seems you're trying to draw an analogy with the double-slit experiment, in which case the electron moves through free space and doesn't see anything except the slits and screen at the end.  In the wire, electrons are constantly running into lots and lots of things, such as atom cores, other electrons, defects, etc.  This interrupts (technical term "dephases") the wave evolution of the electron, effectively making the motion more like classical.  The average distance between scattering events is know as the "mean free path", and in copper this is only a few tens of nm.
For things sized much larger than this, the quantum effects can be ignored, but as things get smaller they can make a difference.
