As you say, there are international treaties (which are nothing to do with the EU) in place to facilitate aircraft of one country flying over other countries.
What I can see happening is that the UK would reciprocate, and withdraw all UK airspace to EI- registered aircraft, and flights to and from the Irish Republic. The UK wouldn't be affected too much, as transatlantic flighs would route south of Ireland, or across Northern Ireland, in UK airspace. Irish flights would only have the option of going west then south, or a long way north before turning east to Europe. That would massively increase their fuel costs and make the routes unprofitable.