AirBnB would be one option, but there are many apartment rental agencies out there catering to tourists. I suggest reviewing Tripadvisor for recommendations and getting the name of several agencies. Homeaway and VRBO come to mind as well as AirBnB, there are also many private rental companies. Since you want to see a number of areas, including some outside of England, you will need to balance the expense of having a Homebase plus spending lodging nights elsewhere vs just travelling from point to point. London is not a cheap city, and if you are travelling, or just spending a month there, it is difficult to get a sense of what living there may be like.
One advantage of London though is that it is the Rail hub for Britain, has quick rail connections to Brussels and Paris (several hours), and has many low cost flights to much of Europe. You can save by cooking for yourself when there, plus many of the National Museums are free.