Equity —formerly British Actors' Equity Association — is the trade union which advances the interests of actors, stage managers and models in the United Kingdom. Until Margaret Thatcher's government outlawed the notion of whole professions requiring union membership, there was no way to work as an actor in Britain without being an Equity member.
Since this abolition occurred in 1988, it can safely be assumed that almost all actors in the 1963 version of Doctor Who belonged to the union. A notable exception was Eddie Kidd, a stunt double brought in on the last serial of the "classic" era. His admittance to the production by John Nathan-Turner's staff caused another stunt performer to walk off the set in protest —though JNT was fully in compliance with both laws and union regulation in effect at the time.
The need to get a union card and the strictures placed on them explains the shape of most Doctor Who actors' career paths.
A conundrum of membership prior to 1988 was that one needed a union card to work in the industry, but that one couldn't get a card without being a working actor. Thus most actors had to obtain work in regional productions, which awarded provisional membership in the union. After enough regional work, actors became eligible for full membership, which could then lead to national work. This is why so many Doctor Who actors' biographies, like that of Elisabeth Sladen, have relatively large sections devoted to time spent in regional theatre or on television programmes that only broadcast in a part of the UK.
Additionally, Equity members are required to have unique professional names. It is for this reason, and not vanity, that some actors have been forced to create "stage names" for themselves. Notably, both "David Tennant" and "Peter Davison" are creations to satisfy Equity rules.