We used a pack and move service about 3.5 years ago for about the same amount of stuff as you, to go 40 mins up the road in London.
From memory, it was about £1200
We didn't get the parking bay outside the new house suspended, so that saved a bit.
We used Ward Thomas, who were absolutely excellent. They sent us a spreadsheet list of furniture, and we just entered how many we had of each item listed (beds, sofas, chairs, wardrobes, books etc etc) and they gave us a quote based on how big a lorry they would need.
The only thing they would't take was the contents of the fridge and freezer (in case we got food poisoning from partially defrosted food)
On the morning of the move, they knocked on the door at 8am, we left with our freezer contents in a coolbox, and our washbags and essentials in another bag, and went to a cafe near the new house.
They packed everything, and got to the new house early afternoon, and we all sat around waiting for the call to say we had completed.
As soon as we got the keys, they were a machine.
All the furniture and boxes were in the right rooms within an hour, and they even put the clothes back in the right wardrobes!
As soon as we had unpacked everything else, we phoned them and they came back to collect the boxes and bubblewrap.
Best money we spent in the whole moving process. I'm never packing myself again!