Basically, don't feel pressured by them 'wanting to move fast' You might be in a position to exchange contracts this time next month maybe if you are lucky, any earlier than you're very lucky to have two lots of solictors on the ball and a council etc conducting searches efficiently. Then after exchange it's 2-4 weeks after for the agreed completion date usually, you may get it quicker if they've already moved all their stuff out and you're in a position to move.
It all just takes time, can get overly stressful because humans are involved, but overall its not that complex a process. I think what grinds my gears is that solicitors can be so shit, but this is all some of them do, every freaking day, the exact same process over and over and they are still, fucking slow, and shit and can't manage to give you good instructions. Although saying that mine was superb, it was my sellers who was god awful. I wonder if my sellers ever did make an official complaint.