Why do they think that organizations who operate completely outside of the UK, accept no money from UK citizens or otherwise do business there, fall within their jurisdiction? I think realistically all that could happen is they tell ISPs to block it.
If that bagel shop will ship bagels to UK soil or accepting money from UK citizens who are currently in UK as 4chan does, it will be subject to UK laws regardless of whether it has legal presence, assets etc. This is not matter of jurisdiction as some of commenters here believe.
UK has limited enforcement options, of course, but it’s their sovereign right to prevent illegal activities and/or punish for them within their capabilities. This is why 4chan lawyers are asking for political response as if UK were a rogue American colony.
ISPs blocking it will mostly work, I think. Most people can't be bothered with a VPN so if 4chan is inaccessible they'll just read/watch something else instead.
I wouldn't put it past them, they've already ran tv news pieces (or maybe it was an ad, I don't watch tv, let alone British tv) basically saying that if you use a VPN, that you're not thinking of the children.
Maybe they will say British citizens use it in foreign countries as their loophole. I don’t agree with it and I’m not a lawyer but I’m just predicting their argument.
They might care about lost advertising dollars, but the UK has no jurisdiction and judgements contrary to the constitution will not be recognized by US courts.
Their reaction puts on record the constitutional/legal barriers, preserves strategic optionality, and possibly preempts escalation that could still cause real headaches. It's strategic, not a sign that they actually think the UK has a case.