Vaccination has never been compulsory in this country, and we've allowed people to refuse reasonable medical treatment for religious reasons (e.g. no blood transfusions for dying JWs who need blood), so a mandatory Covid vaccine doesn't feel particularly compatible with British values.
With that said, it's become very clear in recent years that roughly half the voting population is incapable of sound judgement, so in this hypothetical scenario, I would support some kind of system where you could prove you had had the vaccine (or prove that you were 'exempt' - i.e. a doctor had recommended you did not have it). I'd support exclusion from certain activities based on vaccination status.
No vaccine is a 100% guarantee, and strains mutate all the time. Plus, as I've already indicated, not everyone who would love to get the vaccine will actually be able to get the vaccine for medical reasons. I think to protect people in general, the anti-vaxxers should not be allowed a free pass to mix with those who have taken the vaccine or who are not able to take it.
It's not quite shipping them off to an island, but it's withdrawing some of their creature comforts. I think that's a reasonable compromise.
Of course, in real life, maintaining any kind of secure database not riddled with errors is above Government competence, so it won't happen. But it's a nice idea.