The UK is not in the strongest position when it comes to those negotiations unfortunately. The remaining EU countries have less to lose by denying the UK an open skies agreement (and many other things) than the UK has by doing the same.
Hmm. If it came to playing hardball, the UK could withdraw from the International Air Services Transit Agreement, as Canada did in 1988, and negotiate individual transit agreements with other countries. How inconvenient would it be for EU airlines to be denied overflight of UK airspace on the way to North America?
Anyway, no need to get over-excited - it's only the day after the vote! At least two years to go before actual exit! Lots of time for common sense to prevail. If common sense doesn't prevail - well, we'll know who our friends really are, and who our enemies are.