You can put a condition on your purchase, for example you will only buy if you complete by xx date (maybe 6 weeks) But they don't have to go for it. But may feel pressured to by their estate agent, as to not lose their buyer (you).
If they went for this you would need to get something in writing from your solicitor and signed by them. With penalties put into place for every week the sale goes over the agreed date.
(make sure you have your mortgage offer in place as it can take up to 4-6 weeks in some cases!!)
But as you are not a cash buyer I think it's hard to impose a set date on them as your purchase relies on you selling your flat, which may fall through in 3 weeks because your buyers may pull out. Or other glitches and hitches that hold things up.
Unfortunately estate agents lie. A lot. So we just talked directly to our buyers through our solicitor as it avoids all the angst of trying to guess what the other person is up to.