The commitment made by Labour is to provide an "open access" internet which can be used in exactly the same way it is now, by any company wanting to provide services. There are many places in the world were this happens. Australia, much-beloved by Torie's for its points system for immigration is one, though, just like the points system you will find criticism of it in the country itself.
This is not a new idea. The government builds roads but we choose our cars and where we are going. On a smaller scale, in a block of flats, a service company may charge for providing the basic services for all. How, I wonder, does that differ from paying tax for basic services to be provided?