New nut arrived today so I stuck it in and put it all back together.
It seems to play pretty well, and the electronics sound miles better than the Squier stuff that I'd tried fitting in it.

The only issue.. is... the first fret still. Despite having replaced it again and re-levelling the board, it seems to be a problem. There does look to be some lifting at the second fret, and it's kinda shabby anyway, so I may rip it out and replace it. If I wind up moving the problem to the 3rd fret, I guess I'll go looking for ever larger fret wire and redo both.

The action beyond that is actually pretty good given I'd had it set reasonably low already, and after half an hour noodling, I really couldn't find any issues beyond the sonic joys of the pickups being too close to the strings initially.

Sound wise I'm actually pretty happy with it. A half decent set of pickups seems to make a fair difference.