Can see that the crest of the thread has been damaged in one place or more, but there should be plenty there to hold it on I'd have thought. Can't see how 'out of round' it might be, but it can't be all that very off, as it was on there and was behaving until it didn't. Maybe it was the damaged bit that prevented it coming off in fact, if it was sat there as shards of debris? - in which case, it's good that the headstock spindle thread is still good.
If the chuck was reasonably true for normal ops, it's probably recoverable I'd have thought.
It's not actually that tricky a thing to chase it as described (I'm a totally self-trained geezer with no properly-acquired skills and 300% definitely not a toolmaker even if I flatter myself I can make the odd servcieable weapon!).
If I were doing it I'd use a chunky threading tool owing to the coarseness of 12 tpi, and I'd turn by hand using the lead screw or a handle on the back of the headstock. Very light cuts. You'd find out how out of round it imight be straight away (as long as the jaw side is close to true), because the tool wouldn't bite all the way round to start with.
It wouldn't be hard to get the tool to pick up the thread squarely at that size - it's the small stuff it's hard to get aligned - jiggling the top slide to find the sweet spot in the thread after engaging the leadscrew.
If you do it, don't disconnect the leadscrew until you're done - wind the tool in and out using the leadscrew only, don't touch the top slide, just adjust depth of kiss with the cross slide and retract tool on 'exit' after each run through, before adding a thou or two on the next pass. Unless you have a reliable thread indexing gizmo - I haven't.
That cavity is quite useful in a way because you know when you have run through the thread to full depth.
It would be good if not essential to have another item to hand with the male spindle thread on it, to do regular checks for fit as you go along, as you don't want to have to de-chuck and re-chuck once you're under way if you can help it. And you don't want to take out more than is needed. It's what your model engineer mate would do, I'm pretty sure.
Whatever, Good Luck!
If no go, then the only real option is to replace the chuck, but as we all know, they are quite pricey.