Two major issues here.
The biggest issue here is going to be at youth level. If clubs can't sign kids from outside the UK then their youth teams will be full of English players. It means having a top academy is absolutely vital now. If you replaced all the foreign youth players in the elite academies with English players, would many of our Academy players actually have signed for us? It makes it even more unlikely that any youth players will be able to step up.
The other issue is signing players from abroad. The EU becomes like the rest of the world. Therefore, you're only likely to be able to sign established internationals and the cream of U-23/21 players. I would be surprised if we could have signed Berge, for instance, but it depends on how generous they are being but I suspect a bigger club would have come in. It means that our lack of scouting network is both a benefit and a hindrance. The former because we've not invested heavily in something which now is much less beneficial. But, it will be much harder to sign players from abroad as much greater competition, so those with the greater knowledge may be able to snap players up first.
As with every single thing in football, it benefits the top clubs and the clubs with the most money.