Same things that happens when you have to apply other EU regulations like the "Right to be Forgotten".
What most likely will happen is that there will be a geoblocked EU repository/ies which will provide free access to research material.
Also it's important to note that only research which has been funded with public funds is applicable for this and depending on under which "Open Access" model they'll end up operating under there might be some additional restrictions.
Well it's the question on who this "regulation" is going to target, if it is going to target publishers then it makes sense for publishers that work within the EU to simply allow Open Access from within the EU only, pretty much every content delivery solution has geo-blocking built in, so publishers already have this capability.
If it's going to target institutions/grant recipients then the only thing they need to do is allocate funds to pay for Open Access which is a bit silly and it doesn't solve the issue, if anything it can make it even worse because publishers could then squeeze authors for even more money because now the public is the one who pays the ransom.