I think we're getting a little bogged down here. Let's go back to basics.
I will assume that (1) we all accept that people should only be out and about if they need to. I will further assume that (2) we accept that limiting interpersonal contact as much as possible is a good thing.
Within those assumptions, if stopped by a police officer, is there any need - either in the traditions of British law, or as it stands today. - to prove who you are? No, and because we don't all hold state-issued ID cards, it wouldn't be practical for the police to demand to see your identity papers - you quite legitimately may not have any.
But per assumption (1) we should only be out for a good reason. So if you have some sort of documentation to demonstrate this reason ('Ste Smith is caring for housebound old people') it would make sense to carry it, and something that associates that proof with you (a photo card, driving licence, credit card - whatever you have and reasonably can carry).
And there's still assumption (2). You may be clear of Covid. Are you sure the police officer is? Or maybe vice versa. Either way, you want your interaction with them to be as quick as possible. So do you want a long discussion with a potential carrier, or a quick transaction where the distance from 'Hello, hello, hello' to 'Mind how you go' is a matter of seconds?
I'm not happy about giving up rights like this (and on a personal basis I am worried about how unworried I am, but that's one for me to sort out with myself so that the response based on my values and the response based on evidence and logic match up). But in the current circumstances, where quarantining everyone from everyone else is currently the only solution available, we need to stop standing on our rights. We need to stop looking to enforce the details of the law. We need to see the big picture and do our best to minimise interaction. If you need to be out and can prove why, do the decent thing and carry that proof. If you haven't got proof, try to get it. If you can't do that, he prepared for difficult conversations - but be as nice as possible to the police when you are having those conversations.