The article is about an Aussie campaign. IIRC, Australia already has various measures to encourage vaccination (lower or nil rate of child benefit if you don't).
It's never been a requirement in UK state schools (private schools can set whatever conditions they like), and I think it would be wrong to limit access to education on medical grounds. And I suspect parents would be tempted to lie and it would be hard to check.
It would be simpler to legislate to make vaccination compulsory, but to police such a system would require merging of NHS records with other government records. And given the general horlicks (and data losses) fom ever bigger Govt IT and ID projects I think it would be a massive logistic fail. An unenforceable law would be a bad law.