So this is purely a “You and your friends” problem, right? No one else at the bar cares how your group internally settles who has bought a round and who hasn’t.
ETA: When I first read the OP, I swear I thought that he was saying etiquette required that you buy a round for the whole bar which seemed insane. The “a group orders simultaneously and payers take turns” mechanism is hardly unique to British pubs.
Hopefully you find yourself drinking with a group like the one I have Beer O’Clock with occasionally. Three of us have either very good salaries and/or no wife or young children. We generally buy the first 3 rounds and then some of the younger, less well paid or family encumbered will follow suit. If they don’t all get a round before we go no-one cares.
Once in a while someone will insist on getting the first round because, “I feel like I haven’t bought a round for ages.” This is looked on favorably but since we are no longer having BO’C as regularly as in pre-COVID times, no-one can remember who did or didn’t get a round last time.
In an era where one drink is plenty I’m having a little trouble parsing “the first 3 rounds”. Care to explain that part in more detail? Just kidding [about the explaining, not about the one is plenty].
I think the rounds culture has suffered a bit from the cost of living pinch and significant increase in the price of pints.
If it’s a small group of close friends, then taking turns doing rounds is still the thing. But many other times; if it’s work-adjacent, a social group like from Meetup, if people are joining and leaving at different times…a lot of times IME people will buy their own drinks or maybe only have a rounds system with the one or two people they came with.
This is just a heads up though. It can never be “wrong” to offer to get a round in, and (almost) every recipient of a drink will try to get you one back.
It’s actually not often a problem these days because I don’t drink in bars or pubs much except with my own immediate family, my workmates or clients. And since I am respectively the dad, the boss and the service provider, I am always paying for at least the first round.