I would imagine if we're going to go whole hog down this rabbit hole then such tourists won't be entering the country unless they've got proof of vaccination that they can show on arrival (or possibly departure!). Therefore recognising all the other apps/paperwork/etc is a problem for UK Border Force rather than individual venues within the UK. I think you can see that trend already in the the Red, Amber, Green business that was being discussed a week or so ago in regards to foreign trips for UK nationals. Go to a red country and you have to isolate in a hotel, amber and you're isolating at home, green you're fine (or something along those lines). I can well believe that similar guidelines will be applied to those visiting the UK from such countries. Come from a Red country and it's isolation come what may. Come from an Amber country and you'll be allowed to go about your business if you have an accepted format of vaccine passport you can show on arrival. Etc etc.
So if you're from - let's say New York - then yes, there will be a check at the border.
Later they go to a pub. What do they show to get in the pub? If it is your New York 'Excelsior Pass' (that New York State are currently rolling out), are the pubs going to recognise what that is and therefore whether they should be allowed in or not?
Or if the suggestion is that any foreign-looking app is going to be accepted because there would be the assumption it has already been checked on entry to the country, that gives everyone a very easy workaround to the current proposals