The chaos would be incredible...it's possible we could see a republic again if its backers of the Good Old Cause could sufficiently divorce themselves from Cromwell's legacy, but I doubt it. More likely that William would become titular monarch and a powerful figure (maybe George Monck, or a noble who was a front for him?) would be regent holding real power.
Two important consequences of the latter scenario:
1) An even more dramatic shift of power from monarch to Parliament than the Glorious Revolution in OTL; in fact rather than swerving from Republic to Absolutism to Parliamentary monarchy over 30 years, the country makes a more easy progression from Republic to Parliamentary monarchy.
2) With no other Stuart claimants and no Glorious Revolution, there is no reason to persecute Catholics - in fact after their treatment under the Republic and the reaction against it, treatment of Catholics will probably be almost as fair as under Charles and James. Thus this England, Scotland and Ireland will be more pluralistic, which is good new for the Irish and northern Scots but perhaps less so for the fortunes of English / "British" general culture and supremacy in the long run, given the drivers for the conflicts of the 18th century in OTL.