The extent of CERN's sponsorship of the development of the Web was that Tim Berners-Lee obtained permission from his boss to work on it during work hours. His boss's main reason for giving the permission was that he wanted CERN to start evaluating NeXT, Inc, as one of CERN's suppliers, and development of this Web thing seemed just as good a test of NeXT's marketing claims about the excellence of its development environment as anything else.
Probably the biggest way that CERN helped was by providing a good living for a relatively large population of potential early adopters of the Web. (Before the "discovery" of the internet by the mainstream media in 1993, adoption of the internet was almost completely by very-high-IQ folks.)
ADDED. The much more important point, though, is the fact that when Tim BL started developing the Web, the Internet was a very fertile and welcoming environment for experimentation. If you are on Linux or Mac, take a look at /etc/services and realize each of the port numbers you see listed in that file is an experiment. (Note that most of the least successful experiments might not have made it into that file.) I tend to agree with those who say that if Tim BL hadn't invented the Web, it would not have taken long for someone else to invent something similar (something that allows owners of web sites to create experience on graphical UIs that are more compelling that the plain text that was dominant on the Internet till then).
In discussions like this about which individuals and organizations caused an important technological development, it is important to keep in mind one's reasons for wanting to know. My reason for wanting to know how much of the Web and the Internet was caused by government is that it helps me decide how likely government-employed or government-funded individuals are to contribute to future important innovations.