There's a fairly good historiography on this subject. I would particularly recommend Peter Fryer, Staying Power: The History of Black People in Britain, Paul Gilroy's Ain't No Black in the Union Jack, David Olusoga's Black and British, though none of these deal extensively with Irish communities, so that history may or may not track against this one.

Olusoga's book is the most recent and thus incorporates some of the scholarship written since Fryer and Gilroy published their works in the early 1980s. Olusoga deals more generally with the status of black Britons in colonial territories as well as back in the metropole, so his book has a much bigger scope. All three books, as well work by a number of other scholars, set out to establish that not only have there been people of African descent in the British Isles since antiquity, their presence has been particularly marked since the expansion of British mercantile involvement with the rest of the Atlantic world. Olusoga writes of a brick being thrown through the window of his family's home when he was 14 with a note demanding that he and his brothers be "sent back", reflecting the common view of many white residents of the UK that black British citizens or residents were invariably the result of extremely recent immigration, a belief that simply wasn't true.

In England, free black presence in significant numbers was oldest and most established in the cities most strongly associated with Atlantic commerce, including the slave trade: Bristol, Liverpool, London, and later on, Cardiff, Birmingham, Glasgow and other cities as the Industrial Revolution began in earnest. Many of them were former sailors or the children of sailors. A smaller number were representatives of African rulers or merchants who quietly remained in Britain after arriving there to conduct business, or people who had served in some other role in Atlantic commerce (as translators, for example). Fryer also talks about black performers appearing in London as early as the 1600s. Into the 18th Century, some numbers of people from continental Africa or American possessions were also brought into England, Scotland, Wales and Ireland as slaves to serve as household servants, who might in some cases have settled elsewhere after being manumitted (or having their families manumitted); by the end of the 18th Century, abolitionist religious groups in the UK also recruited former slaves to join their tours and speak out against the slave, with some of these men and women becoming well-known public figures in their own right. (Olaudah Equiano's 1789 memoir was a best-seller.)

The OP asks about the year 1900, and most of the historians who have written about black history in Britain agree that the late 19th and early 20th Century brought about a fairly significant change in the status of residents and citizens of African descent in the UK. Much of Europe, very much including the UK, passed through a major intensification of racist sentiment and ideas during the 19th Century, which in turn were further intensified during the dramatic expansion of the British Empire in Africa, South Asia, and the Pacific during this era. Early modern British residents of African descent were certainly subject to discriminatory treatment, but it was not terribly systematized or strongly felt compared to the period between 1890-1940 (or arguably the period between 1945-1990). Aphra Behn's 1688 work Oronooko, which was then adapted for the stage, had highly complimentary and romantic representations of Africa and Africans in it, and we know of at least one performance where an African translator representing the king of Dahomey who had recently escaped enslavement in the Americas was taken to see the play as an honored guest and peer of his hosts. That sort of thing starts to become pretty hard to imagine at the end of the 19th Century.

Fryer notes in particular the eruption of mass violence against black Britons in Liverpool, Glasgow and elsewhere at the end of World War I. To give you some sense of the size of the community at that point, Fryer states that the black population of Liverpool in 1919 was somewhere between 2,000 to 5,000 at a point where the overall population of the city was near its maximum historic extent of about 800,000. So by no means a large portion of the population, but substantial enough and quite concentrated (and thus targeted by white mobs). By this point, at least some Black British residents and citizens were there also because of imperial associations--former soldiers or colonial civil servants, or people who came to England as employees of former imperial officials (including as servants).

To get a bit more intimately at the OP's question, I'll mention an example that is discussed at length in Ben Shephard's book Kitty and the Prince. The book deals with a stage show put together by a British impresario that was supposed to re-enact the 1893 Matabele War between Cecil Rhodes' "Pioneer Column" and the forces of the Ndebele king Lobengula, as well as exhibit a typical "kraal" that white British audiences could walk through. He brought a number of southern Africans to be part of the performance, some of whom quietly resettled in the UK after the show concluded. One of them was a performer who was said to have been "Peter Lobengula", the son of the former chief. He turned up a few years after the performance having married a white British woman named Kitty Jewell. While that became a scandal in the British newspapers once the stories ran, up to that point, he'd been living with Kitty without it being a big deal. That shows you both that there were black Britons, some of them just going about their business and being regarded as fairly ordinary and that in the new conditions of the early 20th Century that ordinary life could suddenly be made the target of an especially virulent new kind of racism, the sentiments which at a later date gave rise to the idea that Black presence in the UK was just the result of recent lax immigration policies that ought to be reverse.