If Britain had followed Louis XIV would there still be a British monarchy today?
Any answer would obviously involve some degree of speculation; but I believe it is highly doubtful that the British monarchy could have survived an autocrat like Louis XIV. Louis spent extravagantly, waged war with abandon (he presumably stated on his death bed "I have made war too much.") and relentlessly persecuted religious minorities. His extravagant spending led to the sale of newly created titles of nobility (the "nobility of the robe" as opposed to the more traditional "nobility of the sword") which decreased his tax base, as nobility paid no taxes and all titles of nobility were hereditary. Although he kept a tight control on France during his reign, even he saw that the monarchy he created would not last. He famously once stated "apres moi, les deluge" (after me, the flood.)
The English people would never stand for this type tyranny. In fact, when Charles I attempted to rule without Parliament, he lost his head. Charles' excesses were nowhere as blatant or extreme as those of Louis. The British people were proud of and deeply jealous of their "rights of Englishmen" dating to Magna Carta in 1215. Under no circumstances would they tolerate tyranny. For that reason, it is safe to say that had the English people been forced to choose between a flamboyant autocrat and a democracy, the monarchy would soon be gone.