How about add something to the NHS app that shows green after the vaccine, stays red if you haven't had it? To access public transport, shops, flights etc you have to show the green sign on the app.
I doubt proof of vacc would be needed to get on a bus, the London tube etc, but it will be required for air-travel. Several airlines have already said they need tests, and they will almost certainly extend that to include vacc'd people, and inevitably the rest will follow. Will also probably be required by immigration abroad. The NHS App could be used as a certificate but not everybody has it or wants it, so a credit card sized piece of paper with a QR code would probably be needed too.[/QUOTE]
Air travel will be an obvious one. Public transport is something that may have to think about over time, as will shops, pubs etc. If it means a reduction in social distancing and the ability to mix again all of these businesses will be very keen for this.
How they do it? Probably a mixture of options as you suggest. If it's the difference between staying as you are or getting back to life as normal people will likely accept whatever is necessary.