I came across mine not that long ago - it's around somewhere. As Snuffy said, it was buff with a crest and National Identity Card onthe front, and name and address and date of birth inside, and I think you had to sign it.Size as I recall - about the size of an audio-cassette when closed.
No photograph, very easy to forge I imagine, except of course they didn't have photocopiers. I think you were supposed to have it with you and produce it, if the police asked you. That's what would be meant by being asked for your papers.
If a policeman was suspicious of the identity of a vicar he might require a bit more convincing than just an Identity Card. I suppose a ration book could be used as a proof of identity too. And a vicar might well have a driving license, and he'd have certificates from college tucked away somewhere, and correspondence from the bishop.
As for the environment - if it's the study, books, but not paperbacks - except for Penguin Books - with a vicar, especially the green-backed detective-story Penguins, which yiou stikll see sometimes in jumble sales.
Generally vicars still seem to have studies which are much the same as they would have been then, or a lot earlier. Likely to be a college photo, probably the old panoramic type where, if you stood at one end, you could belt round the back and get in the picture twice. Maybe an oar stuck up on the wall?
1940s house, great programme - and a very interesting site as well.