Clearly a local government is going to be less receptive to more digging if high speed service is in place, but it's not a "natural" monopoly if a deal with the government must be made as a prerequisite to provide services. You are also assuming that the "digger" is necessarily the service provider. In my view, local governments are basically seduced by telcos subsidizing or eliminating the up front costs of building infrastructure. In the long run, it's a bad deal.
I would say the monopoly is "natural" if and only if the local government is required to allow anyone (or no one) to dig. Or if it pays for the digging itself.
Ah, yes it is? Someone owns that land, and it's not you. And for that matter, you want to see expensive, wait till you have to convince 10,000 property owners individually to let you trench up their gardens.
A natural monopoly is an economic concept about how high fixed costs makes a single company the most efficient. Regulation of a natural monopoly is secondary to that. A telco might then be a both natural monopoly, due to how the business works, but also have a government-granted monopoly due to licenes etc.
I would say the monopoly is "natural" if and only if the local government is required to allow anyone (or no one) to dig. Or if it pays for the digging itself.