Costco only accepts whatever credit card the Costco credit card currently is. Right now that's Visa, but it used to be American Express before 2016, ironically.

Supposedly this exclusively allows them to negotiate down the fee really low.

It's Mastercard in Japan

https://www.costco.co.jp/GlobalCard

mastercard in canada too

Surely the grand total of Costco’s US credit-card fees can’t amount to more than a percentage-of-a-percentage of all credit-card spending in the country?

...otherwise WalMart and other major retailers (Amazon?) would have partnered with a single credit-card network to benefit too

Walmart and Kroger have had their own disputes with Visa (for one example) and (at least Kroger) started blocking some of their cards at select chains in order to negotiate a better deal. Costco has super tight margins. The the point that when their execs are talking about the price of goods they don't say, "[Item] is $70." No, they say, "[Item] is 69.99." Their culture is about every last penny of the margin. Their money comes from memberships.

Costco UK accepts all three card networks (Amex/Mastercard/Visa). I wonder how their UK division can do this if Costco prefers to do exclusivity deals that allow them to lock in a low fee?

Europe has lower fees by legislative fiat. UK still has these (mostly) due to the majority of these caps existing pre-Brexit.