This is the sort of thing which gets fairly complex, not least because the UK also has to have a dialogue with entities such as the European Union
Und what do you think will be the benefits of such a vaccine passport?
I am asking because there is already (for many decades) a yellow vaccine paper notebook available, which is generally accepted around the world (e.g. on yellow fever).
If you think that there'll be a global vaccine passport one day - forget it! Such a global vaccine passport is only as strong as it's weakest link. In many country you ll be able to get a virtual vaccination* for a couple of dollars. And I assume some corrupt UK-based GPs will get access to this database.
* no real vaccination, just a checkmark in the database