I suppose you could have a system where every hospital in the country offers birth certificates

Wait, isn't this how this basically works already everywhere? The hospitals give you some signed and stamped medical document that states that, indeed, the baby was born, then you bring it to your local registry office, whatever its called, and they give you the state-approved birth certificate based on it. Or is this different in the UK? Anyway, you don't really need blockchain to further automate this system. It's all just paper-pushing, we've been doing it for millenia with different media (be it parchment, paper, telegraph, or e-mail).