There is a game on steam called Universe Sandbox2 that specializes in stuff like that. You can shoot hundreds of planets into the Sun, make Earth orbit Sagittaritus A, and stuff like that. I recommend it to anyone who likes both space and messing with stuff
But to answer your question using Universe Sandbox2:
In the time of a few hours an inner moon very close to the rings (Atlas in most of my runs of the simulation) will eat the rings, giving more gravity to the moon and start a feedback loop. In 10-11 hours the moon will be of a mass very similar to that of Uranus and Neptune, the solar system's ice giants, and many of the ring particles of Saturn begin to fly into empty space or crash into Saturn at the same time. The collisions between the rings and the moon will however slow the orbit of the moon (this is the same thing that makes some gas giants become hot Jupiters as far as i know), making it crash into Saturn in around 1-2 days, and almost all of the ring particles have flown into empty space now. In 2-3 days, all or almost all of the outer moons are thrown into empty space, only a handful of the original moons are still in orbit and Saturn is now at about 160 earth masses. The Saturnian system is now a shadow of its former self.
And keep in mind, all of this happens in a few days.




