Just a quick point as an American living in Denmark, one of the reasons government programs like this work so well is everything is delivered digitally. We have "e-boks" https://en.digst.dk/systems/digital-post/about-the-national-... official government facilitated inboxes so when they need to notify you of vaccinations or whatever else, it arrives to your inbox. And basically 100% of residents use these systems.
Okay, well Ireland has similar vaccination rates, broader childhood vaccination coverage, and no central medical records at all, so while e-boks may assist administration, it's certainly not necessary.
Which is bad, we definitely should have them. Referral data appears to be managed through Healthlink, which may just be a privatised not always used medical record system.