What were the British territories that were mentioned in the Bury the Chains?

If by British territories, you mean British colonies, there are certainly quite a few mentioned in the book.

For example, the British colonies in the Caribbean are mentioned, specifically Barbados. The book tells us that Barbados was the easternmost of all the Caribbean islands and was often the first stop for slave vessels sailing from Africa. During the eighteenth century, European colonizers such as the British brought African slaves over to Barbados to work on...

If by British territories, you mean British colonies, there are certainly quite a few mentioned in the book.

For example, the British colonies in the Caribbean are mentioned, specifically Barbados. The book tells us that Barbados was the easternmost of all the Caribbean islands and was often the first stop for slave vessels sailing from Africa. During the eighteenth century, European colonizers such as the British brought African slaves over to Barbados to work on the sugar cane plantations. The British resorted to importing slaves after the indigenous Indians of the Caribbean succumbed to European diseases such as smallpox and typhus.

Of course, there are also mentions of the thirteen British colonies in America. One colony mentioned is Georgia, which was originally a free state. The author also mentions the British colonies of Jamaica (which became England's largest slave colony), Sierra Leone, and Nigeria. 

In chapter 10, the author discusses the Sierra Leone expedition to resettle former slaves and to bring "Christianity and British influence to Africa." Later in the book, he mentions that the name of the ship sent to explore what would become the British colony of Nigeria was the Wilberforce.