In the same sense that a blockchain can be forked by using software that only accepts certain types of block, is it possible to fork the WWW in a similar manner? e.g. with changes that neuter the ad-mongers.

For example coming up with a way to get rid of these god awful cookies. Maybe ad-monger sites could be allowed in the same way an insecure connection is allowed behind a series of warnings?

The internet is literally just a pipe. There's no limitation binding us to HTTP. You can use any protocol you want over the internet, anything at all.