I have just moved from East to West London. I moved in one weekend and I moved by tube (London Underground).
My boyfriend helped me out and between us we did 4 trips on the underground. In the first I actually forgot the keys to the new flat and ended up sat on a tube platform for an hour while my man went to fetch them!
Pack in advance so you have a good idea of how many things you own - and be ruthless about throwing things away or donating them
Rope in your friends- one more person would have made it all much easier
Be aware of when you need to rest and do so- you won't make much progress if you start snapping at each other
Pizza is a great dinner solution- clichéd but true
Don't overload suitcases and boxes- if you can't carry it up a set of stairs you need to repack! - especially important if moving by public transport
And - most importantly - enjoy your new flat!