IANAL, but I'm confident that the copyright only covers the actual website text, and the list of paths making up the route is non-copyrightable factual information. Reconstructing that information on OSM is the equivalent of a Wikipedia editor consulting a primary source and writing an article in their own words (as indeed they've done for the article about the E2 route: https://en.wikipedia.org/wiki/E2_European_long_distance_path). For the actual location data I've made a point of only using tracks that already exist on OSM, not copying it from any third-party sources.

Nope, signage for the E2 route seems to be pretty much non-existent... I've seen a couple of markers on the eastern sections of the North Downs Way, but that's all. Even documentation of the route on the web is pretty hard to come by - I got the information from here: http://web.archive.org/web/20120212121622/http://www.ramblers.org.uk/info/paths/name/e/e2