Suppose a 1.00 m long wire is connected across a 15.0 V battery.

The battery would produce an electric field inside the wire.

Assuming the conduction charges are electrons, each electron will experience a force acting on it.

Not a very large force but then the mass of an electron is not very big so the electron could and does have a very large acceleration.

If the electron were free to move down the wire unimpeded its velocity when it reached the end of the wire would be quite large.

That is greater than the escape velocity from the Earth. The thermal energy in a wire does give the electrons speed of order of magnitude but their motion is random. This thermal motion goes on even if there is no voltage across the wire.

Needless to say, the electrons are not unimpeded due to collisions and the actual drift velocity of a conduction electron is on the order only a few tents of a mm/s. The energy loss during the collisions is dissipated as heat.