I'll be travelling to GenCon from the UK, will I be able to use my NHS app or a print-out to prove my vaccination status?
One would think so. But you have to realize that they haven't announced any of the procedures yet, so there really isn't any official answer available at this time.
To get an idea what you could expect, you could look at what they were considering doing last year. [They ultimately changed this plan.] However, they had announced that masks were required, but if you showed vaccination proof at a health kiosk, you could get a wristband that made masks optional. So -- maybe -- you're showing your vaccination print-out at a health kiosk. Or something like that. Or at badge check in / will call. *IF* they require vaccines, which is what they are saying now (but could change, I suppose), they will undoubtedly have a solution announced for people like you. As long as you have your proof, I wouldn't be too worried.
But, again, there is no actual answer to this question available right now. The preceding paragraph is just speculation.