Those clubs still exist in London but they're just for the elite to make shady backroom deals with their rich buddies :)
They're really exclusive and they always have been. You and I would not get in, not now and not in the Victorian days. Even 'new money' is usually not ok. You really have to have gone to the right school and have the right family.
Just to challenge that slightly. There is a range of clubs, some are honestly very easy to get into if you end up there for some work event and talk to at least two people. It's the ability to socialize, and lack of clubs focused on new industries that's made them elusive to the new-money (There isn't a National Software Club for example). I'll also knowledge most would run about £1-2k a quarter which is restrictive (by design) cost.
The working class used to have working men’s clubs, but they no longer serve the same purpose.
Which new purpose do you allude to?
Only a guess but they all dropped the "working" and "men" requirements.
The latest incarnation was "Men in Sheds" which eventually got the traditional treatment - https://www.bbc.com/news/articles/cg5qd9l3094o
Hmm yes I've seen men's sheds but they weren't actually men exclusive and very topical to making stuff. A bit like a makerspace but less focused on tech and more on woodworking.