People seems to forget just how badly state run services typically are, even compared to AT&T and the like..
We've had nationalisation and it was a complete shambles. BT had no interest whatsoever in providing Broadband until Telewest came along. You couldn't even buy your own phone or answering machine until 1982 - before then, only a BT engineer was allowed to connect equipment to their network.
The straightforward solution is simply to offer a grant or tax credit to any company that provides the first high-speed connection to a house. Progressively increase the subsidy over a number of months or years to create a Dutch auction between BT, Virgin and whoever else throws their hat in the ring. This would progressively push up the LTV of customers relative to the cost of acquisition, creating the maximum number of connections at the minimum cost to the treasury.
In practice a conservative government will end up selling off and privatizing the company. Subsequent governments will still act like they own it though, because that one company now controls 99% of the country's infrastructure. The share price will then stagnate forever.
Kinda damned if you do and damned if you don't.
Both businesses and consumers NEED good quality infrastructure at an affordable price. Essentially the country's economic growth directly depends on it.
To accomplish that the government should control all fibre, all phone lines, and all cell towers. It should then re-sell access to these things to ISPs/telephone companies.
With this in place you might still buy your cable from Virgin and your mobile phone service from Orange, but Virgin and Orange would be renting space on the government's infrastructure to accomplish it.
Similar to the Network Rail model but less terrible.