In the recent Enola Holmes movies, some characters are played by people-of-color actors who appear to have social standings similar to those of white people. Does this portrayal of non-white people in these movies bear any historical accuracy at all?
Yes. There are historical records of many specific Black people living in The United Kingdom between 1870 and 1930 (or so) who were respected professionals. Medicine and the military seem to be professions where the most records of Black Britons exist.
For example, this image of one page of the 1818 London Gazette shows Nathaniel Wells, Esq., who was Black, was appointed to be Sheriff of Monmouthshire.
David Ochterlony Dyce Sombre was elected MP for Sudbury in 1841 (before being removed for corruption after nine months), decades before the era of the films, and was of Anglo-Indian descent.
More in the era of the films we also see Dadabhai Naoroji who was elected MP for Finsbury Central in 1892, who was born to a Parsi Zoroastrian family in Gujarat.
There was also Benjamin Disraeli who became Prime Minister in 1874 who was born to a Sephardi Jewish family (although he converted to the Church of England as a child). Whilst today he would likely be viewed as white, his Jewish heritage likely would have meant he was not seen as such in his day.
Prior to WW1 the black population of the United Kingdom was much lower, so examples are harder to find, but James McCune Smith was an African American who moved to Glasgow to study for his medical doctorate (which he received in 1837, before returning to Manhattan), during which time he published scientific articles in the London Medical Gazette.
Augustus Merriman-Labor was a Sierra-Leonian lawyer who moved to London in 1904 where he taught sunday school and was Lincoln's Inn (one of the inns of court), although he was only allowed to establish a commercial practice as a barrister in 1909. In his 1909 book railing against the racism that prevented him being called to the bar for five years he claimed London's black population did "not much exceed one hundred" and "to every one, there are over sixty thousand whites". Given the polemic nature of the work it seems likely this is an exaggeration however, and the census did not begin reporting ethnic identity until the 21st century.
I have only seen the firs film where the only PoC characters I recall are Lestrade (played by Adeel Akhtar, of Pakistani-Kenyan descent), and Edith (played by Susan Wokoma, credited as Susie Wokoma, of Nigerian descent). From the above examples, an Asian man rising to the position of detective seems entirely plausible (although would likely be seen as unusual). Edith working in a coffee shop (whilst also running jujutsu classes) also does not seem implausible, although note that the historical figure she is based on, Edith Garrud, was white.