I did the same thing in January, I just rang my current mortgage provider and they did all the maths for me. We sold and bought the same day so although we had arranged bridging finance we never actually used it.
It's easy to arrange that nowadays though, apparently in the past it was a nightmare.
Basically the cheque for the sale is received and lodged and then the bank issues the cheque for the purchase.
Your mortgage adviser will figure out taking all costs into consideration how much you now need to borrow.
I will give one word of advise, leave yourself about 5K leeway as unexpected charges do arise, and it is handy to know you have that cushion in case of emergency. If you don't need it you can always pay it straight off the mortgage.
Good luck.