I would expect a much nastier backlash against Scotland. OTL the worst was limited to the Highlands where the clans had supported the uprising. If the Jacobites enter London and humiliate England, George II and his government might not be so forbearing. Of course that would be a really dumb thing to do in terms of politics.
If the Jacobites get nasty enough in London (we're talking Clansmen here, most of them dirt poor and very angry, and the beautiful town houses, bulging coffers and alluring daughters of London's upper class might just be too much for the fraying bonds of discipline to hold), English opinion might demand brutal vengeance, leaving Scotland in a state of subjection similar to that of Ireland. I doubt it would last, but while it did the economic and cultural damage could be huge.
Imagine: Edinburgh and Glasgow are burned. No Adam Smith, no Gray's Anatomy, no free university tradition - many of the next half-century's finest scientific minds are robbed of their arena. If the Scottish economy suffers from punitive measures, that could have a knock-on effect on the pattern of industrialisation as the northern 'coal towns' can not look to Edinburgh for their models and become more like the home counties (actually, for that it would be enough for the English to feel revulsion at the though ot acting 'Scottish'). Robert Bruce may well become a revolutionary voice of sedition rather than a popular literary star.
Finally, what will happen in Ireland if 'London is taken'? The news would be welcomed, no doubt, but without French help any uprising would very likely stand no chance. Another Battle of the Boyne? Ugh.
Of course, if the Scots limit themselves to plundering Whitehall, the London mob might even join them for a bit of fun...