Thanks for the replies and advice!
It's noisy in all switch positions, but the noise is different.
Adjusting the tone and volume controls changes the noise. It's loudest when the tone controls are between 60% and 90%.
I don't think I've got any bad/cold solder joins. I'll check everything out with a multimeter and also redo anything that looks visibly dodgy.
It's noisier than my other guitars and basses. Not much in a low-gain situation, but very much more noisy than even my tele once the gain's up and/or there's a distortion pedal on.
It's entirely possible I cooked a pot, it wouldn't be the first time I've managed it, although it'd be the first time I've done it in a guitar. A couple of the pots are a tiny bit scratchy and one of the tone pots doesn't travel smoothly. If all else fails I'll just replace them all.
One question that's maybe silly when referring to guitars - Are all the "ground" connections supposed to be the same? As in, does the ground on the output jack connect to the ground on the bridge connect to the ground on the pickup wires connect to the shielding? Or is the shielding supposed to be kept separate from the bridge/pickup/pot/output grounds? Or are the bridge/string/shielding grounds supposed to be kept separate from the pickup/pot/output grounds?
Because I've got them all connected ont he guitar, and I know on some effects units the circuit ground wire connects to the output ring and you get problems if it touches the case or the input ground.
With all that said, I'll pull it back apart tomorrow and check and re-check everything, re-solder anything that's not right, and test it again. If I need to replace the pots, eh, at least they're fairly easy ones to find.