As the European powers carved up Africa, Rwanda’s territory eventually went to Germany.
In the First World War, though, the area became a battle zone between the European powers. But while a British force under Brigadier General Crew advanced south to Bukoba (in Tanzania) on the shores of Lake Victoria, it was the Belgians advancing from the then Belgian Congo to the west who occupied the territory of Rwanda and Burundi, taking Kigali on 6 May 1916. After the war, Belgium retained control of the territories as a League of Nations Mandate. The Belgians did briefly cede Gisaka, a part of Rwanda, to us for incorporation into Tanganikya, before it was reattached to Rwanda in 1924. So we have controlled a bit of Rwanda for a short while.
The British Army sent troops to take part in the United Nations Assistance Mission for Rwanda, as it struggled to deal with the aftermath of the horrifying massacres there in 1994.
Rwanda became a member of the Commonwealth in 2009.