Where can I store luggage for reasonable cost in London?
Some of the railway stations have left luggage offices and if you google you can find which ones and the cost. Whether you will consider the cost reasonable or not I cannot say. Most luggage lockers have a max. time for storage, like 72 hours or something like that.
You only said London. I can tell you that my son stored a footlocker at Heathrow T5 for 3 nights, I forget exact price but it's on LHR's web page. No hassle involved, including retrieving it before his flight home.
Left Luggage has a list of all places to leave luggage in London and the airports.
Last year we had this problem and needed to leave stuff for 4 weeks. Airport was VERY expensive. The hotel we stayed at very kindly kept it for free, even though we were not using them on our return. Most kind of them and saved us a fortune. We would have been cheaper sending it home, as the Gatwick luggage storage place suggested, except there would have been no one to collect it!