Is the electron flow restricted by filament of a bulb? Suppose that the electrons move with a speed of 0.0005 mm/sec in the copper wires. Do the electrons move with the same speed in the tungsten filament as well? This has to be true if current is same in both the wire and tungsten. But the resistance of tungsten is very high in comparison to copper, shouldn't the current decrease in the filament?
 A: 
Suppose that the electrons move with a speed of 0.0005 mm/sec in the copper wires. Do the electrons move with the same speed in the tungsten filament as well? This has to be true if current is same in both the wire and tungsten.

Current and speed of electron are two different things .The relation can be simply stated as
I = e V A n
where I - current , e - electronic charge , A - Cross sectional area , n - free electron density , V - Drift speed of electron.
It is not a requirement for the two to be equal.

But the resistance of tungsten is very high in comparison to copper, shouldn't the current decrease in the filament?

Current will remain same . Equal amount of charges flow through maybe different area with maybe different speed but in the same time. This is to maintain equilibrium inside the resistor .When charge from one end enters the resistor ,equal amount of charge leaves the resistor . If the equal charges have not left the resistor,an additional field will be generated inside it to push an equal amount of charge out .
Finally as I depicted in the above formula ,current and speed of electron are related along with two other quantities. I will not like to make any claims on it as I do not have the required data. But again speed being greater in wire than filament is not a necessity.
Perhaps some of our highly experienced members may provide a data-backed answer.
