Without turning this into a spurious debate about 'what's really inside these vaccines' and similar crap, I am one of those people who has strong reasons to believe they can't have the vaccine due to allergy. (I am awaiting a call from the hospital to carry out tests to ascertain.)
Whenever vaccine passports or similar have been mentioned, the metaphorical small print of the news articles always explains that having taken a recent negative test, and in some cases having had Covid recently, would be just as valid. That's in order to avoid being discriminatory to people like me, to pregnant women, and to people with other conditions.
In my opinion, most of the insistence on 'Covid passports' (which I think is the term the Govt is actually using, as opposed to 'vaccine passports') is aimed at having as many people as possible panic and get the jab, which is the main point of the exercise. The more of us are inoculated, the closer we get to that fabled herd immunity - new, vaccine-resistant strains permitting.
All this caboodle may very well end up being quietly dropped, except perhaps for foreign travel.