If you're not too hung up on the storage place being just around the corner, there's a really cheap but good quality place near Brent Cross, and they do free collection from pretty much anywhere in London. They're about half the price of any of the local ones. I used them last year and they sent round a uniformed guy to help lift & shift. Their site is here
Particularly this section on page 12 as part of the list of banned practices:
18. A wolf in sheep's clothing Falsely claiming or creating the impression that the trader is not acting for purposes relating to his/her trade, business, craft or profession, or falsely representing oneself as a consumer.