For non-brits: Triggers broom is a joke revolving around the ideas of the ship of theseus.

In the show, trigger is a street sweep who has "kept the same broom" for 10 years, only periodically replacing the brush and occasionally replacing the handle.

The joke is of course, that it's no longer the same broom.

https://www.youtube.com/watch?v=56yN2zHtofM

In parts of the US it's "grampa's axe": I've replaced the handle 3 times and the head twice!