Current flights between the UK and the US (and, indeed, all the other EU States) are governed by the "Open Skies" Agreement concluded between the US and the EU in 2007.
Open Skies permits airlines from either side of the pond to operate flights from anywhere in the US to anywhere in the EU (the regulations are a bit more complicated, but this is what everything boils down to, basically). The big issue, here, is that no one ever thought of anyone leaving the EU (or, for that matter, the Union!) and so, no arrangement was ever put in place, should that happen.
One major plank within Open Skies, as far as the UK was concerned, is (was?) free access for US based airlines to London's Heathrow Airport. In days of yore, access to Heathrow as limited to 2 US based airlines but post 2008, everyone had been piling in to Heathrow, assuming they could get the right slots.
Once the UK leaves the EU, then it would appear that Open Skies, as such, dies. This means that either flights would have to continue on a case by case or informal basis or some arrangement would have to be made similar to that of Iceland and Norway. Failing this, a whole new arrangement will have to be made. Presumably, then, access to Heathrow would once again, be up for negotiation assuming, of course US carriers do not think Amsterdam, Brussels, Paris or more likely, Frankfurt are a safer bet, which they could easily so do.
Arrangements with Canada will also need to be negotiated. There are also the neighbouring countries which have been included in the ECAA and individual arrangements will need to made with them, if no multilateral solution is provided by means of the UK's also joining.
Of course, there are the old bilateral treaties but these do not, however, provide for carriers of each country to have full access to the domestic markets of the other, as the current system allows. For each route a UK airline wishes to operate, it will need to ensure it has the appropriate traffic rights, as individual bi-laterals are all different. Since 1976 the airline industry in Europe is, to put it mildly, different.
Significant cross-border airline consolidation around three principal groups Air France/KLM, Lufthansa, and IAG has taken place and low-cost carriers taking advantage of seventh and ninth freedom rights to operate between countries in which they are not licensed, has been rife. In the case of RyanAir and now Easyjet it's: "UK? ... nothing to do with us, Guv..." One wonders if it will have to be the same for British Airways.
As with EU flights, when it comes to negotiating new bi-lateral deals, the UK will have to start from scratch; aviation has moved on too much since 1976. The question is, with the ever increasing flying range of aircraft meaning there is no need to land at Heathrow as "The Gateway to Europe", will many US and other overseas airlines (or countries) bother? If they do bother, then the UK and more specifically, South East UK airports (LHR, LGW etc) will be very much on the back foot.