Was Rwanda and Burundi former British colonies?

Yes, Rwanda and Burundi were both former British colonies.

Rwanda and Burundi were part of the German colony of Deutsch-Ostafrika from 1890 until the end of World War I in 1918, when they were transferred to Belgian control as part of the League of Nations mandate of Ruanda-Urundi.

In 1946, Ruanda-Urundi became a United Nations Trust Territory under Belgian administration. The two countries gained their independence from Belgium in 1962, with Rwanda becoming a republic and Burundi becoming a monarchy.