BoyBrumby
Englishman
It'd probably be touch & go, depending on who you spoke to and whereabouts in the country they were. Rugby doesn't really disappear either, the Six Nations is actually pretty sizeable over here. Since we (England) decided we'd gotten too big for the old British Championships in footy the Six Nations is the main chance for internecine squabbling on the sporting field between the constituent countries of the UK. Club Rugby (both codes) is also a far bigger deal than county cricket too inasmuch as people actually pay to go & see it. CC is really rather like the Loch Ness Monster; often spoken of, but rarely seen.I'm not sure that having the World Cup as a gold standard is such a massive competitive advantage. Most sports which are very fervently followed in a particular country have very traditional competitions that weren't initially started as a World Cup of nations: take the Tour de France or Paris-Roubaix of cycling, Holmenkollen Ski Games, the Ashes, the National and American Leagues of baseball, the Rose Bowl game, Monaco Grand Prix. All were organised reasonably locally and without any international overarching body setting them up. (If it hadn't rained so much in 1912, we'd still be watching Triangular Test series, I reckon.)
Admittedly all those sports are struggling to spread to new countries, but sports with football-style World Cups are merely seen as copycats of football, and do not really captivate a country the way cricket does in India, winter sports in Norway, cycling in Belgium and France. I'd hazard a guess, too, that the amount of people in England who can quote a reasonable world ranking of cricket is larger than those who can quote a reasonable ranking of rugby, simply because rugby appears to vanish between World Cups (like cricket "vanishes" between Ashes series) - feel free to correct me, but it's how it appears from a) this forum, b) BBC website, c) the occasional English paper. Of course those who are interested buy Wisden and Sky anyway.
Lose tradition, and all you get a few people tuning in at World Cups and nothing else.
I'm possibly the wrong person to answer this as I am a Union (& League for that matter) fan. The WC has certainly worked for Union tho inasmuch as it does provide a showcase for the sport of a global stage without really overshadowing the other longer-established elements of the sport like the Six Nations & Lions tours.