Another update: I started it up again today to try more things to see if they're working, mostly electrical things.
The heater does work, so that's good. I thought it didn't work because the heater core was clogged or something.
All of the lights except right parking light works. Bonking the parking light did make it switch on so most likely just a loose connection. This is something I have to fix as it's an instant MOT failure but it's not a big deal.
When going from N to D, the car does lurch forward which is not a good thing. It's not dangerous or an MOT failure, but it's something that does need to be fixed ASAP. I've looked at two different documents from Toyota on this and both say it's most likely a solenoid issue, but it could also be an issue with the gearbox oil filter that just needs to be replaced. I'm hoping it's the latter as that's a much easier thing to do.
Rear suspension needs adjusting. It's far too stiff, even on grass. This shouldn't make it fail the MOT but tbh it should be a fail although there's no way to really test for this. Suspension much stiffer on the rear than the front, especially on a RWD car, makes it prone to oversteer. Not only that, but it has increased danger of having a blow out going through a pothole. Luckily this is relatively easy to fix. I just need to soften the damper settings. It's a shame it's not like the front suspension with super easy access. I need to remove the rear seat again to get to the adjustment knobs.
The horn does sound a bit bad. It might just be a loose connection but it sounds off and a bit quiet.
But other than those things and the things in the above post, it's looking good.