I used Removal Experts for my recent move. They did a lot of things for us – packed up entirely, moved some stuff down to a 2nd house, moved the other contents into storage, then got them back out of storage for us to move them into our new flat, and then they helped us with the unpacking.
I wouldn’t hesitate to recommend them. Despite all of those moves, nothing at all got broken, they were always on time, and seemed to work extremely hard (the longest break they took was about 10 mins for lunch!). The team were always friendly and cheerful, and it made the whole move far less stressful.