A the risk of infuriating a few passionate Scots, they're not going to vote for independence. Seriously, I'm not a gambling man but am genuinely thinking about sticking a sizeable sum on the 'no' campaign winning the fight.
If however some astronomical change were to happen, it wouldn't make any difference to the game. The R&A is a self appointed world body which, with the exception of the USA and Mexico, governs the game globally. Secondly, The Open (noting that it ISN'T The British Open) is not a national championship. Once the R&A run out of excuses and actually take the event to Northern Ireland it'll be heading south of the Irish border before too long.