Two of the major parties support it, but it's not entirely obvious how much public support there is; it's not most people's top issue, and it's easy to make polls say what you want depending on the question you ask.

You'd get different answers if, for instance, you ask "do you want to have to show ID or submit a picture of your face in order to access many sites on the Internet".

I'm not sure there'd be much of a difference because the British people are broadly speaking a rather paternalistic society.

The entire concept of public support breaks down when the majority of the public doesn't actually know what a VPN is.

I sincerely doubt that the average UK (or US, or French, or German) citizen even knows what a VPN is beyond "that thing you sign in to for porn," let alone knows enough about it to have an informed opinion on the laws surrounding them.

Neither does the average politician, so where exactly is the impetus for changing the law going to come from?

I would guess the vast majority of parents support these laws. They are disgusted with the social media platforms who shrug and pretend they are just dumb pipes when it comes to filth, predators, and harmful content, while at the same time keeping users engaged with addictive algorithms and tracking everything every user does and knowing everything about them.