How many different drinks do I need to know to start out as a part time bartender? (UK)?

You should know at least a dozen of the most popular ones like Martini, Old Fashioned, Gin and Bitter or Tonic, Smith& Kerns, and look at the clientele of the bar for a clue to what to learn as far as what is new and popular. Buy a good bar tenders bible at a used book store so you can get a feel for the proportions of booze to mix etc and you will be able to pick up the rest along the way. Unless the establishment has its own recipes for some signature drinks they will not be planning on teaching you any drinks. Go to the bars where you think you might want to work and hang out barside so you can watch the bartenders and waiters and see what is expected.