The UK and all EU countries operate block lists.
If 4chan is found to be supplying content "illegally" (by local definition, not "US-imposed international law"), they can be added to the block list.
Then all credit card, donation, advertising, supply, datacentres, servers, online services, etc. based in the UK... will be forced to stop supplying services to them. ISPs will be forced to block them.
Well, not even forced. ISPs already have to abide by a blocklist maintained by the Internet Watch Foundation (a UK quango). 4chan will just end up being added to that. ISPs will suck down the list once an hour or whatever they're already doing. 4chan will be blocked.
Sure... there'll be ways around it. Of course.
But to the average user, 4chan will be gone from the Internet from the UK. Which ... perfectly achieves what the UK wants. Nobody cares if 4chan goes bust or not, or loses all its money from the loss of advertising - there are plenty of other countries. That's not the aim.
The aim is "this website didn't comply with UK law" - "Oh, look, you can't access it from the UK without having to jump through so many hoops that you basically have to have deliberate intent to do so".
Bam. Aim achieved. And now you can see all the people trying to access a site that's deemed illegal through it deciding of its own accord that it doesn't want to implement age-related measures to a site with content deemed unsuitable without age-verification.
You just put a massive target on all your visitors. You just lost all your UK revenue. And the UK achieved its aim much like any other country that doesn't want you to view a website (the US does it for pirate and emulation ROM sites - and even enlists Microsoft/Google's help to do so!), China does it for all kinds of things, the Middle East countries do it for all kinds of things, every country... has some kind of blocklist like that).
It's like a DMCA takedown. It isn't going to stop them existing. It is going to make it much harder for people to stumble across that content, strip it from search engines, make it harder to use UK services to provide that site (if they do), make it impossible to take UK payments (for advertising or from users), and so on.
It's a really quite significant hurdle. 4chan can laugh it off if they want. Maybe it will have little effect on their site as a whole. That's not the point. That's not why they're doing it. They're not doing it to bring about the downfall of 4chan. They're doing it because the UK have rules about websites, and the websites that don't follow them won't be easily accessible from the UK. That's it. And that's precisely the outcome they'll get.
Honestly, it'll take about 10 minutes to add 4chan to the IWF lists and then every major ISP in the country will automatically start blocking them because... their systems have to. Then they put in a court order to stop people providing service to 4chan (e.g. credit card companies, Paypal - a regulated bank in the UK - , advertisers, server hosts, domain name hosts, etc.) Outcome achieved.
Will you still be able to get to 4chan? Sure. If you want to pay for a VPN or jump through hoops. Nobody ever claimed otherwise. Will anyone care? No. Will other companies that were thinking of not complying with the same laws think twice about just ignoring them? Hell, yes, especially if they have any customers or services running in the UK.