Welcome here, nzbevan!
If you're going to order a replacement 2C210E version ULA or video improvement circuit, I suggest you order a new keyboard membrane while you're at it (well perhaps after checking that a new membrane isn't already fitted!).
Why? Because most likely that membrane has turned to crap after ~30 years.
If it still works, chances are it won't after you've opened the case.

Ordering a replacement together with ULA or video circuit will save on the shipping costs.
If you go for a replacement ULA, make sure to get a heatsink on it! That'll greatly lengthen whatever life is left in that part. Also the
composite video mod is highly recommended in that case.
nzbevan wrote:So either I need to go back really far to a VERY old TV (I'm guessing that means an old B&W CRT one) or upgrade my ULA chip which, it appears, may let me use any TV with a UHF tuner and Coaxial socket.
The one thing going for old B&W CRT is that they virtually come for free these days. So if you have the space for it, and can find one in your area, then yes it's a valid option. The new ULA or video circuit has the advantage that it's easier to connect your ZX81 to
any TV you come across.
Either way, assume that a fix will mean "turn RF output into composite video". No VHF/UHF channel searching, modulator not needed anymore, better picture quality. The only reason I can think of to use a ZX81 with original RF output, is if you've
only got a TV that has RF input and nothing else.

Should be a non-issue these days...