I completed my PPL outside Europe and had to apply for a regular student visa so after UK has left Europe (at least 2 years away) then any European national wanting to complete flying training in UK would probably need a students visa.
This will also work the other way, consider how many UK students go to FTE for training, in future any Brit wanting to go there will have to sort out a visa, it will be interesting to see how their student numbers will be affected.