To the point where I sometimes wonder whether the UK has "bars" at all, of the sort being talked about. Go into any pub in the UK, and it will be mostly full of people in small groups who already know each other and went there to talk to each other. In a few specialised environments (e.g. a university campus during the first few weeks of a new year) it may be more common for strangers to strike up new acquaintances in "bars", but I'm not aware of anywhere where it's a general custom.

It's not the general custom in bars but you can do it. It just requires a thick skin and some not all that high level of social skills. You start a conversation about some random bollocks with someone and if they don't tell you (politely or not) to piss off, you keep going for five to fifteen minutes, say "It was nice meeting you, you want to hang out sometime?" or if the bar is busy enough you go off and repeat the process on some other people and return to the first person later. It is very, very like hitting on strangers in bars and unless you're cool/interesting/attractive/rich the chances of getting anything out of any particular interaction are low. Unless you're very lucky you need to do this quite a lot.

You are breaking social protocol but you're not living in a tribe or a village, you live in a modern anonymous city. Having hundreds of random strangers who do not communicate with each other think you were slightly odd until they forget you (it won't take long) is a small cost for the opportunity to force grow a social circle.

Perhaps in "nightclubs", which I've never been in, but from observation of the queues outside such places, nobody goes to such a place alone, whatever they then do inside.

I have gone to nightclubs alone. This is really unusual because if it goes wrong and only practice makes it not go wrong, on average. But similar advice to the above re:bars applies. This is one of those areas where there's a lot of good stuff in PUA that can be applied to areas other than picking up women.