Prior to the First World War, the franchise (who is allowed to vote) of the United Kingdom was based on the value of property (land) owned by a person, or the value of the property in which a person lived. A person would generally be allowed a vote if they owned land above a certain value, or if they lived in or rented a house which was worth above a certain value, generally higher than if they owned the land. Parliamentary constituencies in the United Kingdom are divided into two types, county and borough. Most rural areas are made up of county constituencies and most urban ones are made up of borough constituencies. This distinction continues today; parties are allowed to spend more money in election campaigns in county constituencies due to the fact that the population is more spread out and campaigning is thus more difficult. In Borough constituencies before the first World War, there were other ways of getting the vote such as being a Freeman of the city.
After the war, the Representation of the People Act 1918 simplified the franchise, extending it to all men over 21 and all women over 30 who either fulfilled a property qualification or had a husband who did. Any soldiers who had served in the war but were under 21 were also given the vote. It was not until the Representation of the People (Equal Franchise) Act of 1928 that the franchise would be equal.
With regards to voting rights for black people in this period, there was no and as far as I am aware had never been a specific racial element of laws regarding who could vote and who could not. However, the United Kingdom did not see significant levels of non-white immigration until after the Second World War. Whilst there were a number of black people in the country before 1918 it's unlikely that many of them would have met the property qualifications to vote. However, in the time that Peaky Blinders is set there would have been a uniform franchise, so any black man who was of the right age would have been allowed to vote.
I hope this helps, please let me know if you have any more questions.
In 1918, universal suffrage for men aged 21 and over, and women aged 30 and over became law, meaning that anyone in the UK that fit this description could legally vote. The character in the show would therefore have had the legal right to vote.
It's worth noting that black people were a small minority in the United Kingdom before around 1950, and while it would have been completely legal for them to vote if they fit the criteria above, it's possible that just as in the US racism and intimidation may have played a role in preventing some black people from casting their votes, even if this wasn't codified in law in the same way as in the United States.
Having said this, British voting rights before 1918 were tied to whether a man owned property and not to his ethnicity, and the colour of the person had no legal bearing on whether they could vote or not. For example, as far back as 1795, an ex-slave named Cesar Picton, freed upon his arrival in England a few decades before, is recorded as being a successful businessman and owning property worth at least £4000 by 1816. Whether he voted or not in his Surrey constituency is unknown, but he would certainly have had the right to do so.