EX, you can pay extra to get your passport rushed. As for how long you can stay there on a passport it depends. If you go over without a visa (like I did) you can only be there for 6 months. If you want to stay longer than 6 months you will need a visa and I think might have to pay taxes (since you will be eligible for their national health care). I know of Student Visas and Work Visas. I am not sure what the limitations are on each so you will have to go through the link I provided to get more info (I am on a crappy wifi signal here and it is about the speed of dialup so I can't look things up fast). As for public transportation: England has a wonderful train system and a great bus system. Only half of my friends over there had a license and less than half of them had their own car as owning a car is very expensive over there. If you are just visiting (or before you move over there) you might want to look at getting a Brit Rail pass (only available to non EU citizens) which allows for flexible and cheap rail travel. As for apartments you will have to start using their lingo to find more (and better) options: You will be looking for a flat to let (as opposed to an apartment to rent).
I may not be the best source of info since I only spent 4 months over there, but let me know if there is anything I can help with.