So a few things to note on this:

This isn't an entirely new idea. BT was privatised in the 1980s, and before that had been run as a government department, and later as part of the Post Office, a public company but still under the control of the Government. At that early stage, the UK had some of the best internet infrastructure in the world, and had plans to roll out FTTH. Then Thatcher became PM, privatised BT, cancelled the fibre rollout plans, and encouraged more competition in the hopes that would improve investment. Well, 30 years later and we still have < 10% FTTH coverage. (Source: https://www.techradar.com/uk/news/world-of-tech/how-the-uk-l...)

Internet infrastructure is in weird state here. Thanks to BT's legacy as a public/nationalised company, they still own the vast majority of infrastructure and cables. This was sort of spun out into a separate company (Openreach) to BT the ISP, but they are widely considered inefficient and disinterested in improving the infrastructure, and they're still a subsidiary of BT. In theory this move was to increase competition, but it hasn't done a lot. There's only one major ISP (Virgin) that truly competes with BT, built their own infrastructure, and ran their own cables, almost every other ISP just leases the lines from Openreach and sells their own services on top of that.

I tend to agree that nationalising Openreach (the infrastructure and cables) is a good idea, to increase internet speeds and coverage across the country - rural areas tend to have very slow speeds as it's not in the interests of BT and Virgin to improve the infrastructure for a handful of customers in that area. I'm not so convinced that providing free high speed fibre internet connections to everyone is a necessity or a good use of government money. I feel like nationalising Openreach but keeping the existing model of ISPs building services on top of the infrastructure makes sense, and public infrastructure would reducing the costs of leasing those lines, a saving that could hopefully be passed on to customers. I'm not sure how this would affect companies like Virgin who do own their own infrastructure though, would they then be competing with the government?

There's of course a privacy concern to the government owning the internet infrastructure too. While the UK government has already had some pretty draconian policies, and they can force ISPs to comply, they've struggled to implement some of their more "ambitious" censorship policies, in part due to it being difficult to get all of the ISPs to come to an agreement on how to implement them. If the government owns the infrastructure, it's easier for them to implement surveillance and censorship at a lower level, harder to circumvent with VPNs and the like. Internet surveillance and censorship are generally policies of the Conservative party, rather than the Labour party who are proposing this, so I don't think there's any malicious intent with this proposal, but if a left wing government nationalises the internet this year, what's to stop a right wing government of the future using this new found control to implement stronger surveillance/censorship in 5 years?

All in all, I'm on the fence about this policy, although I do think some steps in this direction would be positive. And if you've been paying any attention to British politics in the last few years, you'd know there's even more complexities that what I mentioned here.