I’d assume that people are given a vaccination card just like they are for travel vaccinations, I have one somewhere. To enter some countries you’re supposed to have that card handy with your passport.
As for carrying such a card in the UK, I don’t see it happening, hopefully enough people will be vaccinated & immune to make it pointless, but it could always happen if enough people do avoid vaccination.
The main problem that I’m seeing at the moment in such a card is that there is only a single vaccine approved under emergency use, & there are none approved for children & we know that teenagers are being sent home from schools in large numbers. Also, older people who are vaccinated probably aren’t that interested in going to nightclubs, although I’ve not reached that age myself yet & really miss Slimelight.
So, I think too many cons when we should all be concentrating on getting rid of the virus.