This question is a little bit tricky to answer, as many of the nations we have in modern day were not yet established during the British colonial period, or their borders have since changed.
This website includes a helpful and thorough guide to the many places the British Empire colonized, as well as their dates of independence. Some of the place-names are no longer used or have changed their borders, though. At its most extensive, the British Empire was comprised of the territory of at least 59 modern-day nations. This included the Indian subcontinent, the Commonwealth of Australia, Pacific Islands like Fiji, the Commonwealth of Canada, the thirteen American colonies, much of South Africa, Egypt and Sudan, Burma, Borneo, Malaysia, and even parts of Antarctica!