I used James Removals recently (to move me out of London - boo - and into Kent). They weren't the cheapest around, but the service was faultless, and they dismantled and reassembled all my furniture, which no one else would do.
I used local firm RocketVan last week for my move from SE1 to SE18 as I used them 6 years ago when I last moved. They were absolutely fantastic. Robert in the office sorted out the quote and date etc., and my actual movers were Graham and two other guys. Nothing was too much trouble and they were cheerful and professional throughout, even though my old flat was on the 4th floor with no lift! I packed my own stuff (with boxes that they delivered to me a few days before) but I think they do a packing service too, not sure. No connection at all, just a really happy customer.
I've also used Rocket Van for 2 personal moves and for my business - they are fantastic. They do pack as well as move and they do a very good job. Hardworking, friendly and they've never broken or damaged a single thing of ours. I highly recommend them.
They are based in the arches on Union Street so very local.
(again, I'm just a very happy customer - no other connection.)