I think the verification would start at the origin site - wherever the page located. All traffic originating from the client side bundle or on the server would have to stay in the EU.

Cloudflare is going to provide you with a local IP for whatever you are proxying to. This actually makes it relatively simple, Cloudflare would just need an EU checkbox on their proxy. If you enable that you are subject to EU regulations, otherwise the site would not be available in the EU. It would be very easy for Cloudflare to implement such a filter.

In terms of other services (analytics etc), companies would just eventually have to host EU based services (some already do that).