I really really really REALLY wish that when this is discussed a difference could be made between a global system and an internal English system.
Anything to do with international travel with have to fit with an agreed global system. I don't see that happening any time soon and we won't have any choice even if dear Kier does think its "not British" If we want to travel, we will need to comply "end of" as someone else would say. Such a passport based scheme would not work in the UK as not everyone has, or wants, a passport and they cost much moolah.
I believe that its not illegal to refuse entry to shops, pubs so on to people who can't prove they have been vaccinated on the same lines as the "No shoes, no shirt, no service" signs seen in holiday spots and the dress code enforced by hotels and restaurants including the Venice Simplon train. In any case it should be possible to allow the system to accept a medical exemption. it would then be up to individual shops and so on to decide what proof they would accept. It would need to be something that could be used by everyone without incurring significant cost, so no mobile phones or passports.
Its also possible to dismiss people from employment for pretty much no reason at all if they haven't been there for 2 years, similarly its easy to not give work to people who work through agencies or are on zero hours contracts. As I have said before I am against compulsory health treatment of any kind and understand the HR issues, however if my late mother was still here and needing care, I would be off the vaccination for care workers fence faster than you can say "jab"
in any case it will need to wait until all over 18's have been vaccinated.