OK, I desoldered (very carefully) the internal RAM mod and spent the morning going over the ZX81.
I'm satisfied it's clean (no tiny splashes of solder etc.) and the traces are good (checked continuity on every one).
I've resoldered any dry dull bits (there were a couple around the power and the regulator) and, finally, refitted the internal 16K.
Now I just get a white screen. SAVE test doesn't work (although excellent point about the keyboard although I checked that the membrane is good).
I removed and reseated the chips (apart from the ULA) but still the same. I tried a couple of power supplies too (modern and original).
What a shame. It's not end of the world if this is toast, but it's still sad.