What rights does Tim Berners Lee have to the World Wide Web?
Does he have any say over what is allowed on the World Wide Web, its regulations, and so forth? If not, when did he lose these rights and who has dominion over the World Wide Web?
It is mildly hard to say. First, one has to determine if he was an employee hired to do something like invent the WWW. He "spent time" at CERN, but was an independent contractor in his first period (1980), then took up a fellowship in 1984. "Fellowship" is generally not an "employment" relationship in the relevant sense, and academics are rarely "employees" in the common law work-for-hire sense. It is reported that he "began creating the software and standards for the web on his own as an informal project within CERN". So if anyone holds IP rights (to what?), it would be him.
Given where CERN is, it is not self-evident whose copyright law would be applicable. He reports that he wrote the code in CERN Building 31, which appears to be in France, though CERN appears to be in Switzerland (CERN is, in fact, in France and Switzerland). Article 17 of Swiss copyright law says
Where a computer program has been created under an employment contract in the course of discharging professional duties or fulfilling contractual obligations, the employer alone shall be entitled to exercise the exclusive rights of use.
and otherwise, "work for hire" is not an applicable concept under Swiss law. France, which has the same basic droit d'auteur concept, also have the software exception in Intellectual Property Code Article L113-9:
Unless otherwise provided by statutory provision or stipulation, the economic rights in the software and its documentation created by one or more employees in the execution of their duties or following the instructions given by their employer shall be the property of the employer and he exclusively shall be entitled to exercise them.
In either venue there is a very narrow path for CERN to hold the copyright to the original software, one that is not likely to match the facts of Berners-Lee's relationship with CERN.
It is also true that CERN created a version of web software (the usable version), which was released into the public domain Apr. 30 1993 (p1, p2). It is likewise known that he approved of, and propagated the release of the IP that we associate with the WWW into the public domain.
The idea underlying the web is not subject to copyright protection (the text of the proposal would be), but the original code that he wrote would be. It is unknown what relationship exists between that original code any current code, but it highly unlikely that any current code copies original code. Further speculation about the relationship between the original code and anything that exists now would have to be addressed in a History of Computing SE (if there were such a thing).
As for patents, since he did not patent the method, it is ipso facto unprotected and out in the open. From the legal POV, the world wide web is not a single thing, so it is meaningless to ask who has dominion over it.
First, at the time of the invention he was working for CERN, ownership of the IP he created belongs to them.
Second, CERN decided not to patent the invention and released it publicly, so they don't own the IP.
Third, even if they did own the protocols, they could only charge people for using it, they would have no rights over what was transmitted over it.