Well, I'm now getting a little confused. Earlier last week (Monday, I think) we heard that the solicitors had everything in place, apart from the lease information for the upstairs flat, which they're expecting from Land Registry, and once they have that they can exchange.
Later in the week, however, we had to go to the solicitors to give them money "before they can start searches", so had they done them before or not? 
As to the mortgage side of things, the broker sent the application off to the lender on Thursday. We've since heard we need to send in a few more things (signature & direct debit form - little things) and originals of my payslips. When our application gets to the top of the pile (Wednesday, we hope), they might want to see more things, but they're hoping to be able to instruct the surveyor on Thursday. Hopefully take only a few days for that, so hopefully we'll have the full offer by the end of the following week.
How do I stop worrying about getting gazzumped? These few weeks are really tense. We've now spent out money on the process, and vivid imaginations are conjuring up a cash buyer swooping in and taking our gorgeous flat our from under our noses. 