The period between exchange and completion is negotiable.
The seller may be OK with a month, but if there is a chain it's relaly unlikely you'll get the whole chain to agree - the normal period is a week, maybe 2.
You could hand your notice in before you exchange, but this is really inadvisible - if one thing is sure to cause delays and fall throughs it's this, it's just the way of things.
So, probably the lesser of the two evils is to overlap the rental. From there, check with the lender or broker how the lender will deal with your first payment and what they call 'initial interest' - the interest betrween the day you move and the end of that month - they all do it differently and timing is everything - the bills can either fall easily or badly. You'll be able to figure out the optimum date to move. Some lenders will add the initial interest to the loan, a bad idea in the long run, but may solve a short term problem.