Some peers have a super tight wallet where they basically care next to no cash at all, while others always have a healthy sum on their hand. Is there a preferred way to do it?

Yes, but it's different for everyone.

I don't like to have less than £50 in my wallet, and refill from the hole in the wall in lumps of £100 or £200. I use a spreadsheet to keep track of income and expenditure, and project both out month by month for a year ahead, making conservative estimates of everything I can't predict exactly.

This works for me, because (a) I'm well enough off not to need to watch every penny -- the future part of the spreadsheet is a prediction, not a commitment -- and (b) buying or not buying something I see is not influenced by the amount of cash I happen to have in my pocket. I always have cards with me anyway, and I use them for most purchases over 10 or 20 pounds in order to get a record to enter into the spreadsheet, and later verify against a statement. Cash is only for smaller, incidental things and I don't record those transactions.