UK passport holder applying for chinese visa in turkey - where and how?

You can try to make the application from the embassy of China in Turkey.
If you want to apply for a tourist l type, you should first prepare passport and copy of its data page, the application from attached with a passport-style photo, hotel reservation, round trip air tickets, and your residence permit in Turkey, as well as other documents the consular officers consider necessary.
Next, submit the application to the embassy.