I do a zero-based budget monthly, where I precisely account for every dollar coming in and going out. Some of the categories in my budget (anything I buy in person) are designated as "cash" - every paycheck I withdraw enough from the ATM to cover the next two weeks worth of cash categories. These are then distributed into envelopes per category.

All of this leads up to my answer: I carry around personal allowance (budgeted for!) in my wallet at all times, and extra cash pulled from the envelopes when I'm going to buy something that day.

(When I end up using my card for a cash expense, because I didn't anticipate needing to buy something that day, the cash goes into a separate "Return To Bank" envelope. The next time I would withdraw cash, I just take what's in that envelope first, and then withdraw only what I still need. In other words, using my card is merely a loan taken out against my next cash withdrawal.)

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.

I pretty much never carry cash. If I really need it (like for a poker game) I'll stop at the ATM.

Probably the best way to do it is to budget all expenses, allocate some for cash and some for credit, add up all the cash expenses for a month, divide by 4.3, and take out that much cash every week. You could even keep a budget chart with you in your wallet so you know which expenses are for cash and how much.