Why does a light bulb light up instantly First of this is actually not a one question but 2.
I know that the drift velocity of electrons is less than the pace os a snail but it's the Electric field that gives rise to the current instantaneously.
I had the question that how come a field is set up in a wire. 
Regarding this I found these 2 fabulous answers Why doesn't the electric field inside a wire in a circuit fall off with distance from the battery?
and
What happens when we connect a metal wire between the 2 poles of a battery?
1) Based on this I could make out that the field in the wire does not fall off with distance because there will be a build up of electrons in the bends or after a certain distance to increase the field. But I don't think this is absolutely right .
I would like some one to elaborate more on this.
2) Based on this , could some one explain why does the potential falls across a resistor but remains constant in the wire. Everybody explains it in terms of loss of energy as heat leading to p.d. But I would like the explanation in terms of the field picture in sense of build up of charges across resistors like the above links.
Now to the question.
Considering the build up of charges , etc, how does a bulb (resistor) light up instantaneously. It should take a little time to get hot and glow. Is the process so quick or is there physics in it ? I just can't get how the bulb glows instantaneously ?
 A: The answer is deceptively easy: it doesn't glow instantly.  In fact, it takes quite a bit of time for the resistance of the wire to cause enough heating to cause the bulb to glow.  The time it takes is short with respect to what the human eye can see, but when you look at how fast electronic circuits react, it actually takes a very long time.
As for the potential falling across the resistor but remaining constant in the wire, that too is easy to answer: it doesn't.  The potential falls across the wire just like any other resistive element.  Voltage remains constant across an ideal wire, but no such wire exists.
That being said, in many circuits you can get away with pretending your wires are ideal.  The resistance of a wire may be 0.001 ohms.  If that wire is in a circuit with a 10,000 ohm resistor, the effect of the wire is so minuscule that you can almost ignore it.  However, put that some wire across the leads of a lead acid battery, creating a short, and you'll find quickly that you can no longer ignore that small resistance.
As for the build up of fields, that's a bit trickier, because it happens on a ridiculously fast scale.  The fields building up in wires build up on the order of nanoseconds in some cases, and human minds are not very good at intuiting such fast things.  Fields are also full of all sorts of weird complications which probably deserve their own question (or 5 or 10 questions)
