Two things nobody's said yet, both about the new house rather than the old one.
Parking first, because it has a lead time. If there are pay and display bays outside the new place, many councils will suspend one or two for a removal van if you apply in advance. Manchester's, for example, takes domestic removals, wants at least five working days' notice and charges £30 per bay per day, and you can also apply for a dispensation for the van to park on yellow lines. Look up your own council's page this week. If the crew have to carry everything from the end of the street, it takes longer and costs more.
Second, get every firm to come round and look rather than quote from a list. They need to see the loft and the shed to price it properly, and they need to know about the new house too: the stairs, the turns and the doorways on all four floors. In a lot of four-storey townhouses the stairs are narrow, and that's where big furniture gets scraped or doesn't go up at all. Measure your biggest pieces against the stairs now, and sell anything that won't fit along with the rest of the clear-out.
For the expensive furniture, ask each firm what their cover pays out and whether anything needs declaring separately. BAR members have to carry insurance to the association's standard, but the details differ from firm to firm.
And label every box with the floor as well as the room. In a four-storey house, a box on the wrong floor is one you carry up three flights yourself.