Assuming the question is "what if they had stolen all three and they had gotten away with it". As they barely escaped with one, I think a more ambitious theft would have left
Carcharoth with that much more of a stomach ache and
Beren and
Luthien's various body parts strewn across Nargathrond.
If they had by some quirk of fate escaped,
Morgoth's power would certainly have been dimmed, as he would no longer be in a position of defence. There would be no reason to assail him and so the
elves would not seek to do so. However, I am of the opinion that Beren in his diplomacy and kindheartedness would have distributed the gems amongst the
Sons of Feanor,
Thingol and any other friendly party seeking after them.
While this would be seen as a compromise, it would set these nations at each others throats far worse than before. Morgoth's position of initial disadvantage would be reversed, and he may even have recaptured all or some of the jewels, with the death toll massive and the return of the
Vala delayed, maybe for centuries.