There are some huge and crucial differences between the situations in the USA and UK which it would be misleading to ignore (and even more for SA but I don't know enough about it to comment). In the US in the 1960s African Americans made up 10 to 11% of the population, they were a sizable minority with a deep and complex history within the nation. They were subjected not only to intense social and cultural prejudice, but also an entrenched system of legal, political and institutional discrimination, a relationship that had been evolving for centuries.

In 1960s Britain, however, the black population was extremely small, and before 1948 was largely confined to small Afro-Caribbean communities in places like Canning Town, Liverpool or Bristol (i.e surrounding ports which played major roles in the slave trade). In the 1930s (as far as i'm aware) there simply wasn't a black consciousness of itself as a separate group in the same way as there was in the US, and would be in Britain decades later. In any case, they were not a distinctive minority subject to legalized racial discrimination, political disenfranchisement and enforced segregation in the same way as in the US. But beginning in 1948 Britain started encouraging mass immigration from its former colonies to act as a source of cheap labour to rebuild the post-war economy. The rapid expansion of the black and South Asian British population caused tension, animosity and prejudice which expressed itself through extreme racism, open discrimination and violent race-riots. Many found they were discriminated against when searching for housing (ever in short supply in the UK and especially so after the war) and employment. It was through this shared experience of migration and intense hatred that black British people formed a distinctive cultural identity and worked to assert itself. They often looked to and adopted similar tactics to the US civil rights movement (see the Bristol Bus Boycott, for example).

But Afro-Caribbeans were not really fighting for liberation from a system legislative oppression, but complex social and cultural discrimination. Over the 1960s some legislation was brought in to make racial discrimination officially illegal, but as successive government reports and interventions (e.g. the Scarman Report) and decades of violence, abuse and discrimination shows this was simply one chapter in a much larger, slower and more complicated process. In 1968, Enoch Powell, a leading Tory MP, said that immigration would lead to 'rivers of blood' flowing in the streets. In many ways he was right; the 1970s and 80s in Britain was marked by race riots, police brutality and aggressive far-right nationalism. In many ways I would say that this period was much more of a significant part of 'Britain's civil rights movement' than, say, the 1965 Race Relations Act. It was a time of significant identity formation, grassroots political organizing and social and cultural attitude changes. I guess what i'm saying is that Britain's complicated relationship with ethnic minorities is tied to rapid migration in the 1950s, itself the legacy of a distinctive and complicated imperial history and the pervasive racial prejudice which sustained it.

It would be misleading, I think, to suggest that we had a 'civil rights movement' in the 1960s in the same way as Americans did, or that the two phenomena were part of the same historical movement, even if there are obvious similarities and overlap.