God love ya and may the force be with you! LOL
We moved to London in Aug. 2005 and renting with Dogs has been the most challenging thing about our move. So many rentals were not an option for us. We needed a back garden, which eliminated a lot of properties, and then of the ones that do have gardens, we had to find a landlord that allowed dogs, and then we had to pay a non-refundable, higher cleaning fee in order for them to fumigate the premises once we leave. All of this can really influence 1) where you live, and 2) how much you end up paying in rent. I will tell you that the advice we got was to focus on converted flats, because almost all purpose built flats have a no-dogs provision as part of the leasehold so the landlord couldn't allow your dog, even if it is litter trained, as it is a building regulation, not up to the landlord.
It is do-able, but just be prepared to have to make a lot of phone calls, and to be really persistent. If you want to look for a private listing, you could try craigs list, as their website allows you to easily filter for landlords that will allow pets. In the end we got our place through an estate agent - we found one that was willing to work with us. By contrast, when we walked into one agency and said we have dogs, the guy just stared at us and never said another word - he just wanted us to leave! [sigh] In our experience, England is not as "dog-friendly" as we had thought it would be.
Good Luck!