If you're skint and can't afford to replace your passport or apply for a driving licence, then why are you planning on going out which will cost you £15-20? A driving licence only costs £50 so you're around 40% of the way there if you just don't go out.
I know this sounds incredibly boring but this problem is not going to go away until you get a new passport or driving licence. It isn't as though you can go out now and then magically when you want to go out again, you'll have appropriate ID. You WILL get asked for ID, even if you're in a pub or trying to buy something in Tesco so it isn't like you can just avoid the places that ID meaning its more of a long term investment for you to get a new form of ID as quick as you can rather than just putting it off because you don't have the money, yet still going out anyway. Even if you do not plan on going abroad or learning to drive any time soon, you need to replace it because you will get IDed for the considerable future. I'm 23 and certainly do not look 18 and still get IDed... okay, its now only say, 1 time out of 4, but it still happens and until about a year ago I was being IDed pretty much every time I bought alcohol and as I said, I do not look like I'm under 18 and I'd imagine at 5+ years younger than me, you're going to get asked a lot more frequently than I do!
The places the above poster has mentioned will be the most strict on ID in my opinion. They're the key student places in that area of the city and as you'll know, they just ID everyone going through the door regardless of how old you look meaning if you do not have the right ID they will not be letting you in as there are too many under 18s trying to get in those places.