In reality, none of us can really provide firm comment on this. None of us understand what's going to happen to the UK aviation market following Brexit and if it's going to hurt anybody it'll be us British pilots / trainees and residency permits/visas which will come out of it when we are asked to work in EU bases.
From the carriers perspective, the likes of Monarch and Thomson operating your point to point holiday flights would remain relatively unaffected as the basis of their business is on ICAO policies and not related to EASA in respect of EU Cabotage. However, the likes of easyJet of which 30% of their network is EU-EU domestic flights, they will simply split their business up in to different AOCs to avoid disruption, as sourced below.
easyJet plans to establish an Air Operator Certificate (AOC) in another EU member state. This will secure the flying rights of the 30% of our network that remains wholly within and between EU states, excluding the UK
SOURCE:
http://corporate.easyjet.com/~/media...24-01-2017.pdf
As a current or potential pilot for which I fall in to the latter, this of course raises significant concerns as a Brit, particularly given investment in to training. Nevertheless, it's just going to be a waiting game.
Whilst the UK is currently a part of EASA and it's air traffic organisations are also linked with EuroControl, EASA is inherently linked with the European Union and associated members. In wanting a hard Brexit this may well result in us splitting from EASA or it may not. However, once again the uncertainty created here is simply nobody knowing one way or the other, nothing has been confirmed. One can presume its' in the governments interests to try to retain EASA membership in the modern world we live in, but nothing is certain in life. Not even the UK government has confirmed this in their Whitepaper so we shall simply have to wait and see. If we do leave EASA and return to our own regulated aviation market in respect of licencing then it would be only at this point a UK school licence may differ from one on the continent.
In a nutshell, I can foresee that schools in the UK would still be able to provide you training as many of the larger ones are approved education establishments and in theory could sponsor education visas if need be. As an EU citizen (if indeed you are) things would also be much easier for you in terms of rights to live and work in other EU countries so you'd be much more attractive to the airlines if Brexit really does mess us up.
For the specifics I'm afraid we're going to have to see how things pan out from 31st March on wards. I've got my fingers crossed the industry will be unaffected by it but with that said, the UK Brexit Whitepaper gave a pathetic reference to the industry. So, here's hoping.