The neighbour does not want to
pave his side and mine is pretty badly deteriorated due to its being lower
and water pooling on it after heavy rain.

When you pave, you cause the water to go "someplace else" rathern than much
of it just soaking into your driveway.

If you don't want excessive "neighbor trouble" in the future, ensure that
your paving slops to "your side" that than you don't cause the runoff to
damage his property.

As a matter of practicality, the paved portion would have to be 100% on your
side of the line. If he ever comes around and paves his "half" he will
likely (but not necessarily) "fill the gap."

Personally, I'm not a fan of paving driveways. The same things you do to
create a good base for the paving will often "fix" the conditions than made
you want to pave it in the first place.