Many thanks for everyone's replies, especially for the heads-up on the three different kinds of threads that are a close approximation to M5 coarse. The screws were bought new, yes, from a Chinese eBayer, and advertised as M5, but I suppose that's no guarantee of conformity.
Howard, it is definitely not the drill cutting oversize, but thanks for mentioning it.
I have just bought a single BA2 tap off eBay for £2, and will see how it goes. Given the fact that I have another unexpected solution, explained below, it will be interesting if nothing else to have a go with the BA tap.
The unexpected solution is that the screw part of the thumb screw and the knurled nut are in fact "loctited" together.
This wasn't ideal anyway given that the screws are going to be repeatedly heated in use, and it now allows me to disassemble them and use either a known high-temp loctite or [what I'll probably use] silver solder to join a piece of M5 stainless studding in to the original thumb nut.
I've just heated two of the screws to a dull bronze colour and this allowed me to disassemble them easily with the help of mole grips on the screw and a leather-gloved hand on the knurling. Interestingly on both of them the part of the screw that is inserted into the blind nut has a 2.5mm hex recessed into the end. Possibly the hexes were originally intended to allow the screw to be tightened down into the nut, but not wanting to have to trim off the ends afterwards they just inserted them the other way round.
If I'd known the screws were going to be this undersize I'd have just bought the blind nuts rather than pay almost twice as much for the full screws.