Why is the electric field homogenous inside a circuit? In a circuit between to poles of a battery, there is an electric field accelerating the charges in the wire towards lower potencial (but due to resistivity they only travel at drift velocity).
Is there an electric field around the poles of the battery before the circuit is attached, and is there still, after the circuit is connected? And why is the field equally strong everywhere in the wire, no matter the shape?
 A: 
Is there an electric field around the poles of the battery before the circuit is attached, and is there still, after the circuit is connected? 

Yes. However adding the connecting wires is likely to change the distribution of the field.

And why is the field equally strong everywhere in the wire, no matter the shape?

Usually we design our circuits so that the wires don't drop significant voltage. We want the voltage applied to some circuit elements like lamps or transistors, not just heating up a wire. This means we want the voltage to be nearly constant along each piece of wire. 
Since field is the gradient of potential, if we achieve this, then the field will be very small. If the resistance of the wire is uniform along its length, then the potential gradient will be constant along its length, so the electric field strength will also be constant along the wire.
If you had a piece of resistance wire, with different resistivity or different diameter in different sections, then you could set up a situation where the potential gradient (and thus the field) is not constant along a piece of wire.
