Personal recommendations of ones that you've actually used and been happy with, please.

The properties (sell and buy) are in London, although according to https://www.theadvisory.co.uk/conveyancing/conveyancing-solicitors/ in most cases there's absolutely no need for the conveyancer to be local.
(Also according to that source they should be ones that specialise in residential conveyancing or at least have a specialist conveyancing department and operate on ‘fixed fee guarantee’ and ‘no completion, no fee’ bases.)