Let me correct you for you are indeed wrong, at least as far as the Britsh context is concerned.
Prior to the 2nd World war Trade Unions did nothing to challenge racism at work, indeed their members more often took part in it. THerefore the Union movement was segregated along racial lines, for exampole the Coloured Seamen's Union and the Quite active Indian Worker's association.
After the war labour shortages meant active recruitment from abroad, first Polish ex-servicemen and other Europeans, then British and Commonwealth citizens from India Pakistan and thenWest Indies.They were not welcomed by the left or the trade union movement . While the migrant workers did not threaten a single job because of acute labour shortages following the war, the TUC argued during the 1950's and 1960's that black workers did not integrate with white workers. Inded in 1969 the Trade Union Congress opposed anti -discrimination legislation.
Throughout the 50's and 60s trade unions' preference was to, first, keep migrant workers out of the labour market,the colour bar second, keep them out of the union once colour bars became impractical , then illegal, and third, since many became union members, exclude them from the entitled union benefits. It was left to people outside the union movement to put things right
. In Bristol for example a boycott was organised because of the refusal of the Bristol Omnibus Company, supported by the local Transport and General Workers Union (TGWU) branch, to employ Black or Asian bus crews. I would suggest the inflluence was more Rosa Parkes than Karl Marx.THe boycott had the support of promenent Labour politicians including Harold wilson, but not the relevant union , the Transport and General Workers Union, the racist sods.
In May 1965, black workers went on strike at Courtauld's Red Scar Mill, Preston over the management's decision to force Asian workers to work more for less pay, White workers did bugger all. It wad these injustices, injustices that were actually created and sustained by the lefist unions to maintain the privileges of white workers, that led to the first race discrimination legislation.
In May 1972, Pakistani workers in Crepe Sizes Ltd, Nottingham went on strike over working conditions, redundancies and pay. Initially there was no support from the TGWU, but a solidarity committee composed of wives, family, other Asian workers, community activists and the Black Peoples Freedom Movement forced the union to act.
The dispute resulted in management agreeing to union recognition and the reinstatement of the workers that had been made redundant.In October 1972, a strike broke out at Mansfield Hosiery Mills in Loughborough where 500 Asian workers went on strike for higher wages and against the denial of promotion. The National Union of Hosiery and Knitwear Workers supported the wage demand but not the demand for promotion opportunities.
Once again, a solidarity committee was formed from the community which forced the union into backing the strikers by organising an occupation of union offices. When strikes took place in Courtauld's Mill in Mansfield and E.E Jaffee in Nottingham in 1973, the Mansfield Hosiery Strike Committee was there to advise and support them.
Despite some successes, trade union racism was an increasing problem which was demonstrated in 1974 when Asian workers (many of them women) went on strike at Imperial Typewriters in Leicester over racial discrimination and exploitation.
The TGWU refused to back the strike and management colluded with the white workers and local union officials by enlisting the support of the National Front, who attacked strikers at the factory gates. Black strike committees from other disputes in the Midlands, community organisations, the IWA, Birmingham Sikh Temple, and the Birmingham Anti-Racist Committee raised money and supported the strikers who eventually won the dispute. ( Yes lefty trade unionists allied with fascists).
THe union movement did not get it's act together until General Council at the 1978 TUC Congress, Ken Gill, Chair of the Race Relations Advisory Committee and General Secretary of the Technical, Administrative and Supervisory Staffs Union warning against racial prejudice within trade unions, saying that black workers would form their own trade unions if prejudice prevented them from being elected to union posts.
And there you have it. I don't understand this myth of the removal of petty discrimination against black people being an aim of the British left,or at least the white element of it.They actively opposed integration and equality for their fellow comrades simply because of their race, and colluded with management to depress their employment prospects, pay and conditions for decades.
If you want to see a (pisspoor) satire of those days, Spike Milligan in Curry and Chips is close enough.