Picking up a tune on the fly is a skill which can be learned, like any other. It's also one which a lot of session musicians have - perhaps more than you realise, since the best ones will be able to join in so convincingly you may think they already know the tune.

I would suggest it needs the following abilities:

Firstly, complete familiarity with one's instrument, so that you can play a phrase without having to think about the fingering, and in particular without needing the music in front of you.

The ability to identify the key, including any modal variations, and to play the appropriate notes and chords

As I outlined above, the structure of most folk tunes is simple and repetitive to a greater or lesser extent, so it is then largely a matter of recognising these patterns.

Finally, the ability to bluff confidently when there's a phrase you can't quite figure out. Leave that bit to those who do know it, but if you play something which doesn't clash, or even just pause briefly.