'The Wife' says she's using Britannia Group. Our local depot is Appleyards, but all seem to do the same service. Very helpful and quick to answer queries. Actual move is on the 2nd Sept so can let you know how it goes. They weren't the cheapest but other companies had added charges, so came out the same(many companies didn't respond to basic queries). Avuelo is right about Pickfords, their quote was 8 weeks longer to ship and £1k more!
One thing to check with quotes is they give a shipping time, not the actual time it takes to pick up the stuff, and deliver it.
...and they have a habit of the quotes only going to a depot rather than a delivery address.