m0ther - we're in a similar situation ourselves. We knew our vendor was buying a new house that wouldn't be ready until the end of the year, and had agreed a delayed completion when our offer was accepted - this was last Thursday. On Monday of this week at 6pm we got an oh-so-casual call from the EA saying the vendor needed a 6-week exchange. (Good job we ignored the EA's advice to wait 'a couple of months' before applying for our mortgage, then!).
We're chain-free so we're happy to try for this (even though we'll then have a 4-month delay until completion, ugh) but there's a limit to what we can physically do to make this happen.
So far, we've signed all the paperwork with the solicitor to instruct her - actually drove to solicitor's to pick it up, and then drove to drop it back off a couple of hours later, thereby saving at least two days' worth of mailing time - and met very briefly with solicitor to stress that we are keen for a 6-week exchange and could she please submit the searches immediately (typically they like to wait until you have your mortgage offer but given the tight timescale we have to just do this and hope it's not money down the drain if the survey comes back with big problems).
We've also instructed and booked the survey for next week. Our mortgage is agreed subject to valuation (we have a great broker who was all over this as soon as we told him about the rush). Seller's solicitor has sent the draft contract to our solicitor last night, she's promised to start reviewing it today.
So we're now at the end of week 1 (of 6) and at this point, we've done all we can do and we're now dependent on survey/valuation, searches, and solicitors. If it doesn't work out then we know we've done our best and no-one other than a cash buyer (and maybe not even them) could have moved this faster.
I'll keep my fingers crossed for you if you do the same for me...?!