BT's network was sold by the government, it wasn't given away. Shareholders paid for it and the government of the day gave that money away in tax cuts. BT wasn't handed anything, because the network and all the other assets are owned by the shareholders. Your pension fund probably owns part of it.
The cable companies were never publicly owned. The networks have been built out since the early 90's by various private companies (all of whom failed, none of whom ever made a profit) and are now owned by Virgin media. The significant business problem in their case was the dramatic decline in call charges (ironically driven by competition) that ruined their investment cases and meant that they could never recoup their investment. It takes along time to recover a £1000 install when your customer only pays £10 a month and half of that goes to ongoing costs. Even picking up these assets in fire sales at a fraction of their paid-for value NTL and Telewest couldn't make a profit from these networks.
The most significant problem for any business of whatever flavour investing in broadband is that the rate people are prepared to pay (and prices are still falling) don't cover the cost of doing it. The networks we have out there *points* are a result of the low prices we have. The UK has very cheap broadband and telephony, other countries have faster broadband. I'd argue that you can't have both in the same country without huge subsidy from government.
Empirically, if there was a market for high-speed broadband that could be served economically by a new entrant - someone would be serving it. People come along, people try, people go bust. BT and Virgin have huge networks, huge economies of scale and even they can barely make it pay, requiring government handouts to cover all but the most densely populated areas.