Actually the electric field inside the wire isn't the same, at least initially. This is what the skin effect is all about.
When you first apply a potential to a wire, the electric field is a thin shell only on the outside of the wire. This causes current to flow in a thin layer along the outer surface of the wire. Since the conducting material of the wire has some non-zero resistance, this current in the outer shell causes a voltage drop, which means a layer a little further into the wire will "see" the electric field, which causes current deeper into the wire, which causes a electric field deeper into the wire, etc. In steady state, the electric field inside the wire is uniform along the crossection of the wire.
The fact that it takes time before most of the material in a wire is conducting current means that the effective resistance of the wire starts out high and exponentially converges to its DC (steady state) resistance over time. Taking this further, it also means that the wire looks like a different resistance to different frequencies.
For any particular resistivity of the wire material and a particular frequency, a skin depth can be calculated. For external purposes, the wire can be thought of as being a hollow pipe with the thickness of the pipe wall being this skin depth. This can then be used to calculate the apparent resistance at a particular frequency.
This is a real issue, even for frequencies as low as 60 Hz. Power companies are well aware of this, which is one reason electric power cables are not made arbitrarily large. After the radius gets to around the skin depth, making the wire thicker yields less return for the amount of material used. This is why high current 60 Hz transmission lines have multiple cables for each phase (another reason is to decrease corona losses, but that's another topic).
This is one of the advantages of DC power lines. There are disadvantages to DC power transmission too, so it's usually done for long distances where the savings in cable material outweigh the other costs.