I am 19 years old and I'm on a Full time course at college, its actually just 3 days a week though, I've also got a part time job in a hotel, but only doing around 15 hours a week. My mum wants me to move out and I wouldn't be able to afford it on the little money I earn. During the summer I am hoping to pick up more hours at my job but I can't guarantee they will be available. I wouldn't be able to afford renting somewhere, does anyone know what sort of help I could get, and would I be eligible to apply for a council house/flat?

Council housing isn't for single students with no children.
Students share housing all the time, often very cheaply by getting a 3, 4, 5 bedroom, even sharing bedrooms.
Ask your school financial aid office for info on any new aid you might be able to apply for on your own.
When it's time to file taxes, make sure no one else claims you, and you might even get more than you paid out in withholding. In the US students can get an extra refund; I don't know about the UK.

I think it's a bit different here thanks though