We used http://www.fsmithandson.co.uk/
, family run, Croydon based and amazing. Moved 2 streets away from a rental back to the house after renovations in 2 stages. First day they packed the flat, second day moved everything from the flat and the storage into the house. Friendly, smiley and accommodating, nothing was too much trouble and nothing damaged or broken.
We are in Lambeth and have paid for 2 parking bays on the second day, took them 30 mins to move a car parked in 1 space. On the first day, asked a neighbor to park a car next to mine and have given the removal company a visitor parking sticker.
Good luck with the move!