OP the English Civil War (ECW) of the 1640s and American Civil Rights (ACR) of the 1960s are just not comparable, they are two completely different events. The only commonality is the word 'civil'. The ECW is important for understanding the evolution of governance in the UK. The ACR is important for understanding contemporary resistance and popular forces for change, which I would have thought would appeal to you given your interest in working class histories. ACR has had a huge impact on British working class identity and campaigning.
A couple of PPs have mentioned there being a lack of focus on Black British Civil Rights, with only the US figures being taught. This is certainly true from my experiences. Furthermore, I think the distinction that exists in some people's mind's between black and white histories would be less marked if they could learn how black 20th century campaigning has benefitted the whole of society.
Decolonising the curriculum is not about removing certain subjects in favour of others. It is about teaching all subjects in a rounded manner. For example, when I was at school we learnt about WW2 in a completely Eurocentric way. The colonies were not mentioned at all. When the teacher spoke about 'we' she meant white British people and their European and US allies. We thought the only black troops were a handful of American GIs! Nothing could be further from the truth.
A Tudor specialist explained once how a focus away from the exploits of royalty towards the lived experiences of the working people would allow for much greater inclusion of minorities, of whom there were several thousand residing in England at the time.
When studying Marxism, include key figures from the British Black Power movements of the 1960s for whom Marxist theories were key, and who took a keen interest in working class liberation not just black liberation. Study the life and work of Claudia Jones who was instrumental in establishing the first Notting Hill Carnival to promote peace and unity in an area fractured by race riots. She, incidentally, is buried right next to Karl Marx!
This type of approach promotes a better sense of the convergence of our shared histories, and encourages a better sense of unity in current times, and is therefore extremely important in our society today.