Old UK banknotes, made of 'paper', had no real durability issue.

I think the key benefit of polymer notes is the opportunity for far more security features. Holograms, Prismatic effects, braille printing, variable transparency, etc.

Also, they're probably cheaper, considering most of Europe uses polymer notes too.

The UK had issues with banknote forgery, so a change over to a new base material makes a good clean cutoff to say "no paper notes are valid anymore", which cuts off forgers too.

The US also has major note forgery issues, but to my knowledge they have never taken serious action against fake notes. If they did, they would probably switch to polymer too.