That is one possibility. Things like that have happened in many newly independent countries, usually where there aren't stable constitutional and democratic traditions: Algeria, Zimbabwe, Ghana, etc.
The other possibility is that the party system in a newly independent country may move back to the center. It's no longer a fight against foreign bureaucracy or foreign money or foreign media. People start to think of the government, the economy, and the media as their own. So, for example, after Ireland got self-rule a lot of the radicalism dissipated, and a century later they look a lot like Britain.
Something similar happened with our own revolution. It wasn't government or taxes per se that we were revolting against, but a government overseas imposing taxes on us without our consent. When we got independence and representation, we taxed ourselves more than the British dared to.
So there's at least the possibility that an independent Scotland will develop a viable conservative (if not Conservative) party and government in the future when politics is no longer a matter of getting back at the Sassenachs.
How long has Scotland been slipping away from conservatism and into socialism? I’ve read that Thatcher got about 34% of the Scottish vote in 1979 (with Tory proportion of the vote gradually declining with each election), and I’ve also read that this decline started before Thatcher. I know that Scotland didn’t like monetarism due to its refusal to subsidize Scotland’s dying manufacturing industry, however, which paints a different light.
Americans revolted against their government because of what it did do (and you could say they will soon revolt because of its actions), and that seems to be the case for Ireland as well (but the radicalized elements of the IRA had enormous influence from 1922 to 1969, so I wouldn’t use Ireland as an example of a country which began to slide away from radicalism once it gained its independence). The Scottish “yes” voters, on the other hand, are complaining about Westminster inaction, such as its refusal to make college free and alleged privatization of the NHS. It is true that the Scottish nationalists adopt anti-tax rhetoric, making their platform supply-side (low taxes, high spending), but the result of the adoption of supply side economics eventually leads to either higher taxes or collapse, since politicians will never cut government spending if left to their own devices (and the welfare state has been consistently popular in Scotland).
Because of that, I have a hard time believing that Scotland will be willing to get rid of their welfare state. A yes vote could be a valuable learning experience for them, but it would also be painful. If the Scottish nationalist parties were right wing, independence would be clearly good for Scotland (since Westminster is indeed Leftist and a lover of high taxes), but the nationalist movements there have been left wing and there’s nothing other than the rantings of a few paranoid leftists to suggest that they will change.