I haven't seen the movie, but black Britons is a topic I find interesting, so I can try to give you an answer.

Looking at the IMDB cast list, the black character seems to be Sir Cecil Syers. Now, this is a guy who seems somewhat hard to track down. Best I can figure, he was a South African who served as a diplomat for the British before and after the Second World War. His picture is hard to track down - this is the best one I could find. Syers is second from the left. So it appears that particular character was a bit of creative casting, and that Syers wasn't black.

Now, to address the rest of your question. I'll use numbers and stories from around the beginning of the 20th century.

There were black Britons present in British society before the Second World War. The black population of Britain is estimated at around 20,000 individuals in 1919, with most of them residing in port cities such as London, Liverpool, and Cardiff. In areas such as Tiger Bay in Cardiff, or Canning Town in London, there were communities where black and white people lived side-by-side, and mixed-race relationships were not uncommon.

However, not all of them were dock workers! We know of some black people who worked as doctors and lawyers in the Edwardian period. Others worked as merchants. Some of these individuals were wealthy Africans and West Indians who came to Britain for their education, while others were born in Britain. There certainly was a small but active black bourgeoisie in pre-WW2 Britain. Arthur Conan Doyle even wrote a mystery about a 'black doctor', originally from Argentina, who settled and practiced in Scotland.

As far as black politicians go, England's first non-white MP seems to be John Stewart, elected to the House of Commons in 1832. Including Stewart, there were three black MP's elected before the 1980s. Most of these men were part black, and the descendants of West Indian planters. England's first South Asian MP was elected in 1892. However, Britain's first black cabinet member wasn't appointed until 2002, and the first South Asian cabinet member was appointed in 2014 - so it did take some time for people of colour to gain entry into the highest levels of government.

Of course, it's also true that racism was alive and well in pre-war Britain. I mentioned above that mixed-race relationships were accepted in areas like Canning Town or Tiger Bay, but non-white people could face significant prejudice. White dockworkers sometimes apparently refused to work alongside non-white workers, and even 'well-educated' black people were sometimes refused service at establishments in London. In addition, although black people were present in some areas, in other areas of the country, people had never seen a black person before the early 20th century. Early 20th century England was certainly not a race-blind society, nor a multicultural one by modern standards.

But all this to say that there was a not-insignificant black population in Britain in the early 20th century, and it's not unfeasible that some of the wealthier members of that population could have ended up in some sort of government position by 1939.

Sources:

Black Edwardians: Black people in Britain, 1901-1914, Jeffrey Green

Staying Power: The History of Black People in Britain, Peter Fryer

Before the Windrush: Race Relations in Twentieth-century Liverpool, John Belchem

"Interraciality in Early Twentieth Century Britain: Challenging Traditional Conceptualisations through Accounts of ‘Ordinariness’", Chamion Caballero, Genealogy 2019, 3(2), 21.