Do you mean first-claim? My first-claim membership has always been independent of any racing, even though in practice that's the main time you'd cite it. While a member of three clubs, I clearly understood that I could only be first-claim at one of them, even if I never raced.
I'd expect to enter any event in the name of my first-claim club, and to wear its colours. I often wear them even for Audax, just out of loyalty. However, it's more that than rules. I don't think that time trialling has ever insisted on club colours. With British Cycling races, including CX, I'd have expected to enter in the name of my club and wear club colours.
Of course, it's possible now to be a member of a club that's not registered with BC. That was rare, years ago, because clubs always registered with the controlling bodies if those clubs had racing members.