You do realise that in practice, there isn't all that much of the network that was built with public money ? And what was publicly funded was SOLD to BT's shareholders on privatisation (whether the price was right is a different discussion).
The network as it is now is a very different beast from what was flogged off 3 decades ago. Yes they had a head start in that there was an existing networks of ducts, poles etc - but that's been expanded a lot since then.
There really isn't a viable model for competing "last mile" networks. You don't have two different companies digging up your street to offer electricity, or gas, or water, or drainage - that would just be madness. Not to mention, if taken to extreme, you'd have a choice of two (or more) different road networks to get to your house !
That's the reason we don't (for the majority of us) have such end connection competition. It's costs thousands of pounds per mile to dig up roads to lay a network of ducts. You have to put that infrastructure in place before you can connect a single customer. And then you have to persuade enough customers to switch to your service to repay all those costs (and loan interest). Meanwhile, your upper price on the service is more or less set by BT who have the economy of scale from having a network that's been built of many decades.
When you look around, you tend to see that alternative networks fall into about 3 categories :
1) There's Virgin Media with it's cable network which stands no chance of being extended into lots of low density areas. But that wasn't built by VM, it was bought for pennies in the pound from the liquidators of the many cable companies that started up, incurred the cost of building out the network, but just couldn't get enough return to pay back their loans and investors.
2) There's small specialists that service places BT won't - often on a "if X sign up now, we'll come and service you".
3) And there's "community projects" like B4RN which rely a lot of donated labour (ie volunteers) and favourable treatment from landowners to keep the install costs down to something affordable. Something like that project doesn't work in even small urban areas as the costs go up very considerably when the network has to go into public roads rather than (mostly) under someone's field.
It's notable that B4RN found a number of cases where they announced a plan to extend coverage to somewhere BT had refused to service - only to find BT "suddenly finding that it was now economic" and would service that area as a spoiling tactic. Other altnets have also reported the same problem.