There would be some sort of Parliamentary Settlement as 1689 has already happened, without two disinterested Hannoverians, the devolution of power into Parliamentary hands won't be so quick. It'll happen over the 18th century slowly; even with the Hannoverians it took until Victoria for her position to be weakened, as Melbourne, in Parliamentary interests, informed her of what her rights were and she listened, compared to meddling George IV and William, both of whom paid little mind to Parliamentary majorities and appointed who they wished even if Parliament opposed.
But the crown would be in a strong position; I wouldn't say an absolute monarchy, but a sort of pseudo-absolute state in that Parliament is necessary will continue to exist and sit, but the King's will still be able to play a vital role; much like George III did, I imagine the Stuarts playing a role in rewarding supporters with bribes and gifts and being able to bring down bills or even governments with enough opposition.
The crown may be considerably poorer though, as we may see no Civil List as we know it. It was created for George III in exchange for giving up the hereditary crown lands. I've always imagined a restored Stuart Monarchy reviving the Star Chamber to act as a strong arm to remove from the teeth from Parliament as well.
In terms of foreign policy, definitely pro-French. I agree with Thande and these restored Stuarts would abandon the hopes of Charles II and James II for religious toleration for Catholics. Protestant Dissenters received toleration in '89, but I imagine as Thande said the Stuarts will laud the Established Church and hold it up on an altar: one they don't pray at, but one which they respect as the Establish Religion in Britain. I imagine if we see a religious reawakening in the 1720s as IOTL, the Methodists may have an interesting time if the monarchy is hostile towards them.
With the American colonies, I don't see any change from the benign neglect. James II's centralist attempts were unpopular and I don't see them being revived. Being neutral or friendly with France will also reduce the need for defense expenditure, although I'm sure colonists will still distrust the French. They'll probably still be upset if, say, Paris leans on London to stop illegal settlement in French Louisiana. A sort of royal proclamation line, but not exactly.