We have just gone through our house sale and had all the same questions as yourself. It took 11 weeks from acceptance of the offer until completion and like you we were trying to guestimate when it would all be finalised. We did ask the solicitors for an indication of timescales but of course these can never be guaranteed. We told nobody of our emigration plans.
We wanted to apply for OAA so ending up taking a gamble & booking flights. The ones we booked could be changed for a small fee (75)- Singapore Airlines Flexi fare- but of course we had to provide these flight numbers and date of arrival in Adelaide to the OAA people. We put a buffer of a week in Singapore which would allow us to change our flights & leave the UK a week later without arriving in Adelaide any later. We applied for OAA with just over 6 weeks to go & the email back from OAA people states if you change your dates then your OAA application will be cancelled.
We asked for 2 weeks between exchange & completion on our house sale, agreed by both sides. In the end we got 4 days!!!! We had booked the movers & the exchange kept getting put back day after day, but the movers - John Masons were great & hung on until the last minute for us. We had zero days to spare handing in our notice to use the original flight dates we booked. Looking forward to a week in Singapore now. Its been 2 years since our last holiday (unless you count a week in Scotland)

All in all it was a very difficult & stressful time, never knowing until the last minute if the house sale would go through. Would have been much easier moving from somewhere rented in the UK.

Whilst we would really like OAA, we have always looked at it as a bonus if we got it & budgeted for not getting it.

Good luck with everything. You do mention you have job offers to take up so at least you'll be earning when you get there which I am sure will cushion the blow if OAA doesn't work out.