Belgium will ban anonymous SIM-cards.
Does this ban apply to foreign SIM-cards? For example, can I use my anonymous UK SIM-card in Belgium, or will I be refused access?
No! You'll be able to use your UK SIM card just fine. As an example, I've been using my prepaid T-Mobile SIM (US) in Belgium with no problems.
It's illegal to sell a SIM-card anonymously (this applies to stores) and it's illegal to have non-registered SIM-cards (this applies to providers).
It's up to the providers to make sure that no-one can use a SIM-card of them without being registered. Some let the stores register the SIM-card and the users info. Others made an app where the users can register themselves.
So only the Belgian providers and the SIM-cards they distributed have to be bound to a verified person.
This link provides some additional info (it is in Dutch).