If the Syrians had broken through the lines on the Golan
They did break through, at least in the southern Golan.
and threatened to rampage into the Galilee, Israel would have gone nuclear. Bet on it.
Assuming Israel was fast mobilizing, I doubt the Syrians would've entered Galilee or the Israelis would've needed nukes if they did. The Syrians wanted to reach the Jordan river, or the '67 border, in part because it would've made their best defense line--their best hope of holding out against large Israeli reserves. Going farther would've played into Israel's hands, both politically and militarily. It would've undermined the arab claim that they only sought return of their own territories.