Why do metals conduct electricity faster than heat? I have read this question:
https://physics.stackexchange.com/a/561125/132371
where cuevobat says:

You will note if you examine the tables on conductors that some metals that are good electrical conductors also conduct heat well for the same reason - heat is also a vibration of electrons and depends on free electrons to easily transmit heat.

This says that (the conduction of) both heat and electricity depend on free electrons.
But we know that metals conduct electricity close to the speed of light.
Now if I have a metal wire, and heat one end of the wire (various ways, for example put it into hot water), the other end will heat up very slowly. Yet if you connect one end of the wire to electricity (outlet), electricity will flow through it and reach the other end (if you connect something to it, like you touch it) with speeds close to the speed of light.
What causes this difference between the speeds of the propagation of those two phenomena (electricity and heat) in metals?
Just to clarify, both depend on free electrons, so why don't both propagate at comparable speeds?
So far, I have two very interesting answers, and we got so far to the point where I see that in the case of:

*

*electricity, the electrons "push" each other, and this transfer of momentum propagates close to the speed of light




*heat, the electrons transfer energy (hot electrons colliding with cold electrons), and this is much slower


So now basically, the question comes down to why is transfer of momentum faster then transfer of energy between the free electrons?
Question:

*

*Why do metals conduct electricity faster than heat?

 A: In the case of heat transfer, the wire stays electrically neutral, so one electron can only interact with other electrons in it's close vicinity by collisions, but has no effect on electrons far away. So the signal about the increased temperature (i.e. higher thermal velocity) travels by collisions from one electron to the next. In the best case scenario the random collsions happen such that the next electron always travels in the right direction. Then the information about the increased temperature can travel at most with a speed equivalent to the electrons thermal velocity, which is about 100 km/s or 0.3% of the speed of light.
In the case of electricity, you push additional electrons in one side of the wire. This charge surplus will cause an electric field to form which travels at the speed of light. So this electric field can almost instantaneously affect all electrons in the wire, even electrons far away on the other end of the wire. All electrons will start moving in the same direction driven by this electric field.
So in short, heat transfer is analogous to sending a message through a relay race, while electricity transfer is equivalent to cheating in that race by calling the person on the end of the line with your phone and telling him the message.
A: 
Why do metals conduct electricity faster than heat?

If by electricity you mean the flow of current, then the question only makes sense (at least to me) under transient conditions with respect to how quickly steady heat flow is establish vs how quickly steady current flow is established, as discussed below. Under steady state conditions it's an apples and oranges comparison since current is the rate of charge transport and heat is the rate of energy transfer.
Transient Conditions:
Take a metal conductor, a wire. At time t=0 you apply a voltage difference between the the ends of the wire. An electric field is almost instantaneously established and electrons almost instantaneously begin moving throughout the conductor with some average drift velocity. So current is almost instantaneous throughout the conductor.
Take the same wire initially at room temperature throughout. The mobile electrons in the wire will have the same random thermal motion throughout the wire roughly proportional to the temperature. Now at time t=0 establish contact between one end of the wire with a high temperature constant temperature source with the other end in contact with a constant lower temperature equal to the room temperature. Thermally insulate the circumference of the wire to prevent heat transfer to the surrounding air.
The difference in temperature between the ends is analogous to the potential difference. The random thermal motion of the electrons near the high temperature end will increase. Through collisions with the electrons farther away from the hot end increased thermal motion will progress towards the lower temperature end until a linear temperature gradient is theoretically established along the length of the conductor. However, unlike the situation for current, this progression of thermal motion will not be instantaneous as in the case of the collective motion of charge. It will take time.
Steady State Conditions:
The applicable steady heat flow equation is
$$\dot Q=\frac {k_{t}A(T_{H}-T_{L})}{L}$$
The applicable current flow equation is
$$I=\frac {k_{e}A(V_{H}-V_{L})}{L}$$
The equations are roughly analogous with rate of heat transfer $\dot Q$ analogous to rate of charge transport $I$, thermal conductivity $k_{t}$ analogous to electrical conductivity $k_e$, and temperature difference $T_{H}-T_{L}$ analogous to potential difference $V_{H}-V_{L}$. The length and cross sectional area of the wire being $L$ and $A$, respectively.
But current and rate of heat transfer are different things, so it's comparing apples to oranges.

"However, unlike the situation for current, this progression of
thermal motion will not be instantaneous as in the case of the
collective motion of charge. It will take time." Can you please
elaborate on this a little, why this is, this is key to my question.

First, the motion of electrons in the case of current flow is collective motion, called drift velocity, which is proportional to current. That motion is due to the unidirectional electric force applied by the electric field to the electrons. That electric field travels in the conductor near the speed of light. So all the electrons immediately start moving.
On the other hand, the thermal motion of electrons when the conductor is heated is random. They don't collectively move along the conductor. Electrons with high thermal motion (random velocities) near the heat source collide with nearby electrons away from the source having less thermal motion (lower temperature) transferring kinetic energy to those electrons raising the temperature of conductor further from the source. They in turn collide with electrons nearby them and so forth until the thermal motion of all the electrons in the conductor has increased raising the temperature. All this takes time.
Check out the following video showing how the temperature of a heated conductor slowly increases along the length of the conductor as evidenced by color of the conductor.
https://www.youtube.com/watch?v=y-ptY0YG9RI
Hope this helps.
A: Sippose  you have hosepipe full of water, and turn on the tap. You  see that  water comes out of the nozzle  end of the pipe almost immediately. The "push" from the water at the tap end reaches the water at the nozzle  end at the speed of sound in water ---  more than 1,000 feet per second.  The water entering the pipe at the tap takes much longer to reach the far end.   It travels at a  few feet per second.  If the water in the pipe was cold, and it is a hot tap that you have connected, you have to wait the longer time for the hot water to reach the nozzle. In a wire the "speed of sound" for  the electrons is the speed of light. At a current of one ampere the  electrons drift at a few inches per second. If there is no actual current, the heat travel only by hot electrons hitting cold electrons, and  the heat travels  even slower.
