We don't have an independent system. The UK's emergency alert system supports local areas. For example, during a storm in January, Cornwall got an alert, but not London. Plymouth once got its own alert.

However, during storm Éowyn, the first ministers for Northern Ireland appeared to request it, announcing it in an emergency press conference in the afternoon, alert hitting in the evening. It began with:

Issued by the UK and NI Governments

There's also alerts prefixed by just the "Issued by the UK Government", "Issued by the Environment Agency", and the Plymouth one was "Issued by Plymouth City Council" .

So I think local authorities can issue them using the national system. There's probably a civil servant in Whitehall it has to pass through first. Don't want some angry councillor giving off about dog poo ¯\_(ツ)_/¯

This fella here has a pretty decent explanation for a short video

https://youtu.be/eOuiMEZ5vng