If you sign contracts on the house you are selling, and have agreed a leaving date, well tough. You'll have to move out and rent somewhere in the meantime. It depends on the people moving in...can they wait another 2, 4, or 6 weeks or longer?
If they are not in any hurry (which would be odd) you could try to 'rent' your old house from them. I have heard of that working but usually people buying a house want to get in ASAP. You can throw a couple of delays in towards the end but you still have to be reasonable to your buyers.
The chances of a new build going over expected time is fairly high to be honest.
The days of selling one house and moving into the new house the same day are pretty much gone, unless a lot of planets align in your favour. So I would be thinking of Plan B, where you have to move out but have several weeks before you move into the new place.