Anon. because the people I "ran away" from Google search my username.
I have no idea how safe it is sleeping on the streets anywhere, so I can't comment on that, but living in a house/flat in London is safe, it's just fear mongering to suggest otherwise. If you don't know London well you should take as much care you would when moving anywhere new though.
Leaving home with no place to go is unwise, but if you do find yourself with nowhere to live you can contact Shelter (they have a useful website too.)
I was in a very similar situation with my parents and the location miles from anywhere, but luckily I have a decent amount of money saved from previous jobs. I told my parents I was going to visit friends in London for a day, came here, looked at the flats I'd arranged to see from Gumtree.com (very useful -- try the short-term lets section) and chose the first one I saw as it was the cheapest and available from that night. I went back to my parents place on the train, then packed my stuff that evening (quietly!) and left their house at about 4:45 and started walking to the nearest village. At 5:00, I realised my bags were too heavy for me to carry them the 1.5miles to the bus stop, so I called a taxi (hooray, 118118!) which took me to the station in the nearest city. I got to London very, very tired but without any other problems.
I haven't got a job yet, but I have enough money saved that this is OK -- I'm waiting for the graduate schemes to take applications, and working out where I'm going to live long-term.
I wouldn't recommend "running away" if you can avoid it; myself I felt I couldn't avoid it. My parents were upset, but then I was unhappy living with them.
If I had my own place I'd offer you the couch, but I haven't sorted one out yet.
By the way, I know a few people who cycled more than 20 miles each way to work every day, so it is possible -- it's tough, but might be a better option than moving out before you've got some money saved?
Good luck, whatever you decide.