Private GPs tend to be more amenable, as you are paying them. However, they shouldn't really give you a sick note if they haven't seen your symptoms.
However, it sounds like this is just a part of your problem. Travelling through central London is difficult and expensive, and it doesn't sound like your accomadation is ideal. I think it would be a good idea to talk to your personal tutor about all of your issues, and explain why there have been problems with your attendance. They will want to be suportive, and help you attend as much as possible.
Is there any way you can move closer to your uni? I think you will definately need to do so for next year.