Not sure how this will go.
Yes, it is the biggest act of censorship for centuries, comparable to the Licensing Act of 1737 that imposed state censorship on the stage. That's what governments do - control and suppress their citizens. Shepherds and sheep. Most democratic freedom is illusory at best. The courts only function for the wealthy. Super-Injunctions and D-notices etc. Your greatest enemy is always your own government, because they are the control freaks who feel that it is their duty to control and limit what you do. That's why most people don't vote for politicians, but against the ones they fear and hate the most. But don't assume that having the right to do something means they will.
The government want to win votes from the mumsnet lobby. They also want to be able to tap GAFA for very large sums of money (more easily and more frequently than they already do).
There are enough back doors in OSs, ISPs, software and VPNs for state spooks to check out anyone they want to already, and they don't need it to be legal.
Most people using social media (or even adult websites) have never seen cheese pizza on them. Hate is not new. Celebs' agents used to block the green crayon letters. Having it online makes it easy to trace. Driving it underground will make it more dangerous (and certainly won't reduce it - it is a human problem not a tech problem). The BBC have been running internet scare stories for decades to support this eventual censorship and these restrictions on the net, but unless you go looking for this sort of stuff, it is quite rare that you will see it in mainstream surfing.
I'd be more concerned about other stuff that may be buried in the legalese - licensing websites, rating them like movies, banning distributed services, age/ID checking, blocking foreign content and cross border data flows etc. Governments want internet services to stop at their national borders, and want to end Web 2.0 (unrestricted commentary on their corruption). These are things to worry more about.
Governments do like to give themselves God-like powers so they can do anything they want. They rarely have the competence or cash to actually do it all, 24/7. They pick targets for a reason.
I'd expect they would use the powers to take down some wretch every now and again to warn others off, and to bag a few hundred million from GAFA.
If the EU and UK start to regularly tap GAFA for free money, Washington might not be pleased. And if services pull out of the UK/EU, aside from voters hating their governments even more than they already do, the NSA lose the global access that global use of US net services gives them.
The content scanning will already be happening, from online services to HDDs. Giving it legal status doesn't make much difference. You have no privacy anyway. Western governments take the Chinese line: no secrets from the state in the name of public safety and good order.
But the government have broken Britain offline with Brexit - 25% off sterling, one-third poorer, short of workers, services like the NHS, local councils and railways collapsing. Maybe this is the companion piece to break Britain online too. If you break something, it is easier to control. The solution to that may be to emigrate and go somewhere nicer. Unfortunately, other governments will be planning something similar. The model for government today is taken from Orwell and beta tested by China. After decades of globalisation, low prices, freedom to travel and work and the empowerment that the global internet gave us, the Empire may now be striking back. The good times may be over, permanently. The bad guys won. I guess we shall see.