Do many people in the UK feel an affinity with people from Commonwealth countries? Maybe nothing major, but some sort of minor thing?

I don't think so (from what I gathered from my Brit friends & while living there).

As a Brit being totally honest, I do not even have a clue which countries are in the commonwealth...

Nope, none at all.

The question can be asked in reverse maybe.

I can say that I feel more affinity towards Brits than born-and-bred Canucks, despite of living over 20 years in Calgary (and spending only a grand-total of maybe 20 calendar-days--including airport layovers--in UK).

I've actually made the comment that I would be more culturally-compatible with a woman from Doncaster (where I've never actually been) than with one from Atlanta (GA, lived there 6 years, 1994-1996, 1997-2001) or Edmonton (in the same province as Calgary).