I use one "chord" in St James Infirmary that is not in any book I have ever seen but it had the notes I wanted!

That's fine as far as I'm concerned! When I want to sing a song I haven't done before, I'll work out an arrangement that suits me. I won't necessarily use the same chords as the person I'm copying it from. I might transpose it into a different key, possibly for the reason I find it easier to get the effect I want in that key; or maybe simply because it's easier to sing and/or play in that key. I'll just use the original performance or recording as a guide to the chord structure of the song. I wouldn't want to slavishly copy the original anyway, and prefer to apply my own ideas to the song.

Given a key, the basic chords are bleedin' obvious (aren't they?), though of course it does help to first work out what tuning the performer you are trying to copy is in. And then just experiment.

Yes, often the basic chords are obvious to you and me and others, countess, but I think the point of this thread is that they're not at all obvious to some people, which is why we're discussing it.

As for the tuning, I don't play much in tunings other than standard; so often, if the original artist uses a non-standard one, I'll use the nearest standard tuning equivalent (possibly transposed as above).

From the responses so far, it seems there may be an element of experience coming into it, but it could be partly due to a natural ability. It may be the same reason why some people can play by ear and some don't seem to be able to.