The drysuit thing is the only reliable one. Every time I have had a student rent one they have been asked for a card and I have to show my instructor card as they are being trained. Of course you can buy a suit without a card.In the UK, it’s not common to rent a drysuit.
You will need proof of Nitrox training to get Nitrox fills.
I have been asked to prove that I had an AOW for a dive under OW limit.
I never heard of someone asking you a drift dive card here.
The thing is that some of these cards don’t have an equivalent specialty training for BSAC
Inland sites are most likely to ask for cards. Boats don’t. You can do a 80m trimix dive without showing anyone a card. Generally, everyone is assumed to be an adult.
If anyone attempted to show a drift diving card to a skipper, I think the skipper would assume it was home made and part of an elaborate joke.