Another way to go is DAE Options. I do not know if this is availible to US based members, as I am a European-based member, but I suspect it may be. DAE Options provides non-timeshare properties as trades on a ''cash plus exchange deposit'' basis. The owners provide lots of weeks in their property, as opposed to the vacation property owners who deposit a week or a handful of weeks into the regular exchange system.
There is one property in London that is handled through DAE Options, known as 66 The Lexington. It is a 2BR condo in a good central London location, and they often have scattered weeks there availible. Right now,they are only through May, but a call may get you something later. They are priced at one exchange credit (which can be any timeshare or vacation property) plus GBP 494.
The owner of 66 The Lexington has in the past also given some weeks into the regular exchange system at DAE, and at least one Tugger has reported getting it as a straight exchange. I would not count on that however for this property.