Vaccine passports seem fairly straightforward to me.
For the purposes of international travel - the UK is going to have to make them voluntarily available to UK residents, end of. The existing WHO International Health Regulations 2005 certificate is more than acceptable for this purpose, has been issued in the UK for decades, and is between £0 and £15 depending where you get one from - see the blank version
here. A lot of travellers to Africa and parts of Asia will already have one. It requires your passport travel document to be written or printed on the front, meaning that the government (or a shop, business etc) probably couldn't require you to use this sort of document domestically, as if you don't travel internationally, you will not hold a passport in the first place.
Exceptionally unlikely (virtually nil chance) of the UK government(s) mandating and legislating for a compulsory passport, or a law requiring businesses to request to see one as a compulsory condition of entry, although they may allow people to not be tested for COVID on production of some sort of vaccination proof. (Although I suspect mandatory tests to enter high capacity places like stadiums and concerts will be required regardless).
I suspect
some businesses, although probably not the 'big' ones like supermarkets, clothes shops etc will do their own thing and require proof of vaccination as a condition of entry. However, I suspect that won't last too long, and such silly requirements will soon go away on their own accord.
So, in essence, the UK needs to make vaccination certificates, passports etc optionally available on a voluntary position, otherwise international travel will become difficult or impossible to countries to have them as a mandatory condition of entry.