Sorry to say but, for this to happen, the UK/EU Relationship would have to breakdown to such a point where cooperation of any form would be impossible.
Never mind who wins and/or who loses. Look at Iran, Russia, etc. They have a much worse relationship with the EU and the world and yet, Aeroflot, fro example, is operating out of LHR AND LGW and is member of the Star Alliance. Both Russian and European Flights use their respective airspaces regardless to the fact that the EU has imposed several sanctions to Russia.
There will be several losers (certainly no winners) on this but it is such beyond far fetched that I am surprised Ryanair even considered to come with this statement.
As hard as a Brexit could be, for this to happen you would need something in the level of the UK or the EU pulling out of the UN and NATO to even consider.
Some of what you describe is covered by basic ICAO regulations, so would still apply. But most is organised by bilateral agreements. With Iran, Russia and all the other examples EU has working agreements. And your examples are still far from the single market that we currently still have. With the current schedule after two years the EU rules and legislation don't work anymore for the UK. Bilaterals don't exist.
The problem here is time and workload. With Iran, Russia and all the other examples we have working agreements. So we have two years from now to get to a new agreement between EU and UK, have it signed & ratified by both sides (and in the EU case sometimes by the different member states). For one agreement this would be ambitious, but could be done. But there is a zillion agreements to be done at the same time, covering each aspect of the relationship, not only aviation. So I understand if airline officials start to worry and try to get their issues high on the priority list for negotiations.
If no agreement can be reached after the two years, no extension given for EU rules and no new agreement reached, what would happen?
And some other interesting thing: The problem might no only exist between EU and UK. What with the bilaterals between UK and third parties? I understand that UK is not covered anymore by agreements between EU and these third parties, so UK has to negotiate its own agreement with Russia, Iran and others. This means they can not only focus on negotiations with EU, they have to worry about the rest of the world as well. Again, major problem will be time and workload.