Just quickly wanted to mention that hardly any of the BW moorings in London are residential. Little Venice was mentioned, but when we got our new moorings bill, (which included prices and details of all moorings in London), we were surprised to see that it isn't residential. (Although I doubt whether this is enforced). There are only two or three BW moorings in London that are truly residential and they want unbelievably high rental fees - for a sixty footer you could be looking at over seven grand a year.
Have you thought of Virginia Currier? They sell boats with residential moorings, but beware, if you are looking at getting a boat mortgage, they will only lend the value of the boat (i.e. not the overinflated London mooring included price for a boat), so you can be looking at finding tens of thousands of pounds extra, for a boat from the 1970's thats worth at the most 25 grand and needs loads of work.
It's very true what Anhar says, the only way to buy a boat in London and get a mooring is to buy it on the mooring and before you hand over any money, get it IN WRITING that they will transfer the mooring to you.
If you don't do this you could be looking for a mooring for years. We realised that this has been made worse recently - there are some private residential moorings at Three Mills Island, but everyone has to move for 18 months whilst they improve the river there (they plan to transport building materials by barge, to construct the Olympic village and stadiums), It seems that BW has not been letting newly available moorings (we have two empty spots on our mooring), but reserving them instead for these liveaboards to rent whilst the building work is being done. This must be having an impact on the waiting lists.
Finally, if you do want to rent a BW mooring, bear in mind the River Lea is a lot cheaper than the GU (about 40% cheaper) and is nice and quiet, especially further out of London.