As it stands, and it could change, “proof” of vaccination is not relevant. The requirement is to provide negative tests. This makes sense, at present, because the vaccine doesn’t stop you transmitting the virus. It might lessen the transmission but it doesn’t stop it. The point of the vaccine at the moment is to lessen the severity of the symptoms if you have it
In some ways this could be viewed as making the situation more dangerous in the short term. You catch the virus after vaccination, you have mild symptoms and think you are OK because you have been vaccinated so you go out and about, or travel to another country, spreading the virus as you go....
I presume this will change as immunity increases within society but at the moment it’s the negative test that matters. Despite what many people seem to think a vaccination is not a “get out of jail free card” just yet.
Until there is some level of consistency in approach to vaccine paper work, with at least some level of security built in, I think countries will struggle to use them as formal proof. How is the person at the border supposed to know that the “proof” issued by your vaccination provider is authentic if 100s, probably 1000s, of forms of proof exist? As most just seem to be printed bits of card they are also child’s play to forge making them potentially even more worth less.
Any way if anyone wants the full detail it’s here. You should also be aware that rules etc can differ in each of the home nations, eg England, Scotland, Wales, NI.