Do British people say "brr" when they're cold? If so, how do they pronounce the R?

Tbh, I don't think you really understand how the non-rhotic accent works. In this case, the /r/ would be fully pronounced, as it would be at the start of a word. Say bread, elongate the r and skip the ed part and you have what it sounds like.

If you're very used to hearing the bunched r, the British version still might sound softer, but even in the USA (where most people use bunched r) it's still common to hear an r made with the tip of the tongue behind the teeth (upper or lower).

I'm ignoring the other r sounds, but you do find a lot of them across the various regional English accents.

We kind of go "brur".

I don’t have the answer but I’ve always thought it was interesting that British-English spells the filler words that Americans normally write as “uhh” or “um” as “er” or “erm”

Now I'm just imagining a Scotsman pronouncing it and really rolling that R 😂

Poorly

If it is particularly taters out it can be done by clenching your teeth, lips together but not firmly closed, saying 'buh' and blowing. The 'rrrr' is produced by your lips flapping together. It is quite fun to do

Not usually, I wouldn't, but if I would I would pronounce the r like short oo with vocal fry

I'd have thought different regions have their own way of saying they're cold - for example up north, I'm "nithered" is a reasonably common way to say you're cold...