If James II were to leave a son that was raised as a Protestant, it might lead to the Stuarts keeping their throne longer than OTL. This James III & VIII would likely be the choice of Parliament to suceed James II & VII if/once he pursues the policies that lead to his overthrow.
If the Stuart stay on the throne, Franco-British relations might not drop like they did with William III OTL. All will depend on how James III and Louis XIV go along of course and of how James III sees the Sun King's policies.