C680 flight crew reported ATC issued a traffic alert followed by an immediate turn instruction to avoid a collision with an aircraft that descended below its assigned altitude.

Narrative

Cleared for takeoff runway heading to 3000 feet; we were leveling at 3000 feet on runway heading past from Tower to Departure about 40 seconds before this. Departure Control said there was traffic in our 12 o'clock descending and leveling at 4000 feet. I looked down at my Primary Flight Display (PFD) and saw the TCAS Target 1000 feet above us then it continued to descend with a down arrow and was 500 feet above us about the same time I was expecting to get an RA The Departure Controller in a frantic voice; said Aircraft X turn right immediately!" I already had my hands on the yoke expecting an RA; I pressed and held the Control Wheel Steering (CWS) button and rolled into a 40 to 45° right bank due to the urgency with which the Controller said to turn right. There were clouds around us with convective activity to our right as I turned. We entered a cloud bank with a lot of uplift and gained about 600 feet the Controller now regaining his composure; told us to climb to 8000 feet And cleared us back on course tour next waypoint. Not 30 seconds after heading to this waypoint; we requested right of course for convective build at the waypoint the Controller again in an uneasy voice who seemed still destabilized by what just happened just said whoever's requesting right of course "just do it".We never received a resolution advisory based on the proximity and the closure rate I believe we should have; and not sure why we did not. But obviously; the Controller was shaken by what was about to happen.Suggestions - I do not think there's much more we could have done I was ready on the controls as I saw the target descend below It's assigned altitude ready to to do whatever is necessary. My First Officer and I talked about this on our way over the next two hours to ZZZ1; and believe it came down to the regional airliner busted his altitude in the descent and the Controller noticed it at the last second telling us to turn immediately."