Environmental niches filled by squirrels, mice, small dogs, foxes, coyotes, raccoons, hawks, eagles, and numerous others would be left wide open. Those rats would start evolving into any empty niches, that were close enough. Over time/generations the further niches would become available. Smaller rats would take advantage of food sources that the mice used to eat. Better climbers would evolve to take advantages of what squirrels used to eat. Rats that specialize in eating insects would start to specialize.
Some rats would even start to specialize in eating other rats. Other rats would specialize in not getting eaten and fill the niche left by rabbits.
The rats that evolved into squirrel like mammals might specialize in jumping ever further to avoid the ground where the rat-dogs roam looking for rat-squirrels to eat. Said jumping might even evolve into flying squirrel like rats. Given enough time active flight would evolve. Rats would even start to populate the oceans.. much like whales did.
Evolving into a bird requires efficient lungs, light weight (things like hollow bones), and not wasting weight on things like powerful legs, thick skin, etc. But every step of the way would be better for some niches.