At the time the Germans were using the enigma machine what were the British using to encrypt their messages and was it ever broken?
The British had a similar machine known as the Typex machine. The Wikipedia article is quite in depth, and of particular relevance is the section on "Security and usage" that says:
Although a British test cryptanalytic attack made considerable progress, the results were not as significant as against the Enigma, due to the increased complexity of the system and the low levels of traffic.
The Crypto Museum has a list of British cipher machines, although I'll admit I've only heard of the Typex, and I don't know what date the others are from.