We are lucky in that we sold our house earlier this year and now live in rented as we knew we would be moving. So no house selling worries plus we get a rental reference :D
We were the same as you. As each stage went in we had the next ready so could move along quite quick. Our agent advise waiting about 6 weeks after application went in to book medicals and we tried to time the police checks to be back at about the same sort of time.
Will keep you updated :D