Think the month gap is a very good idea, we had it and it made a huge difference!
Wasn't our choice, to be honest, but it does give us some grace (and the mortgage payment is tax deductible...)
Just had a call from the purchase solicitors saying the funds have come through from our mortgage company but they haven't had the funds through from the remortgage. Spoke to the remortgage solicitors (not my choice to instruct them - it was a condition of the remortgage) and they've got the funds from the lender, so that's good, but they haven't sent it off yet and don't know when it will be. It's going to be a long wait but at least the money is out there!