I have been with Andrews & Arnold ISP (AAISP) for some years now, starting with "pure ADSL and then moved to FTTC.
A few weeks ago I moved from central London (E1) to Croydon (CR0). This involved a (BT) phone number and line change too. No problem - I shot off an email to AAISP asking how to best arrange this and got an answer a few hours later.
I don't think I've ever had so much guidance and email-tracking on any online service :) so jolly well done to Michael B. and the support guys for making this as smooth as could be.
By linking the BT order's SIM reference (used to be the FORN) AAISP got my circuit live a minute or so after the BT Openworld chap set it up.
It's been said before, this ISP is not the cheapest BUT their technical service provision and A1 support is what sets them apart from the rest. If you want an ISP which speaks technical, can provide technical services and can be contacted via phone, email or IRC to help the technical user then you can't IMHO go wrong with AAISP.