Credit to your Mum - she's got a point!
Unless you have a habit of sleeping in your fridge you'll want a wetsuit... Water is warmish by UK standards in September as it has been blessed by sunshine on its travels to our shores... A 3/2 would do the job although if you're a complete beginner you're probably as well heading to one of the surf schools and booking lessons and kit hire for your stay... Might seem pricey, but the instructors will get you to your feet quicker than trial and error and if you ding / tear anything kit wise you just give it back, smile innocently and say "Oops!"
It's difficult to accessorise round a wetsuit too, so it's probably a good idea to get the Bug before spending your money on all the kit and deciding you cup of tea is a little different!