As someone who lives in Belgium where it is more rainy than the UK apparently, you just need 120€ of good biking gear (maybe 150 now with inflation)

I have some cheap Columbia rain pants, some cheap rain covers for shoes, a decent marmot raincoat, and a very cheap cover for my helmet. I bike 40-45 minutes each way to work.

I arrive to work dryer like that when it is very rainy, especially my feet, than when I have to park in a parking garage and walk 10-15 minutes. With an umbrella and raincoat.

I mean, it definitely isn't fun at all biking in that, but unless it is really a downpour, it is not crazy.