Epic E1000 pilot reported ATC did not approve their weather deviation request which resulted in them encountering Wind Shear and an uncontrolled 3000 ft. climb. Pilot regained control of aircraft and returned to assigned altitude.

Narrative

Adverse Clearance Revision without timely modification resulted in encounter with severe convective vertical wind shear. Departed ZZZ on IFR clearance to ZZZ1 at 33000 ft. via the ZZZZZ1 arrival. Preflight briefing showed route of flight through convective outlook area. Upon handoff to center; and to my dismay; ATC cleared us only to FL190 and said to expect no higher (14;000 feet below cleared flight plan route). With lots of deep moist convection in sight ahead I asked for higher and was denied. It was necessary to request multiple lateral deviations to avoid buildups. Other aircraft were doing the same. Repeated request for higher and was told airspace above too congested though we could see very few aircraft on TCAS within 40 miles of our position. A 40 degree right turn from ZZZZZ to ZZZ [VOR] had us pointed directly into fast rising convective clouds - towering cumulus. Again asked for immediate climb or turn back to the left to avoid weather. No response from ATC. We entered the cloud and experienced an uncommanded ascent above our selected altitude which the autopilot could not correct. There was intense convective vertical wind shear in excess of 6000 feet per minute. The plane experienced a 5000+FPM instantaneous rate of ascent and inside of 32 seconds had ascended nearly 3;000 feet; far beyond the climb capability of the airplane. I disconnected the autopilot and began roll and pitch changes to arrest the climb; get the aircraft under control and back down to its assigned altitude. The downward descent rate was nearly as high; exceeding 4000 FPM. Within the ensuring minute we were back down to 19000 ft; out of the clouds and headed toward ZZZ [VOR]. But the route ahead looked worse despite ATC saying they could see no weather. Not a surprise because the developing thunderstorm clouds had not reached full maturation. Our datalink weather display was showing severe weather along our route ahead. So I asked again for higher and; when handed off to the next controller; requested vectors to the north. We were cleared to FL230 and were given vectors all the way to ZZZ1. The remainder of the flight was uneventful. The cause of the adverse weather encounter was a result of a controller at workload saturation managing congested airspace full of threatening weather who was unable to provide a timely deviation clearance. My deviation request came too late to avoid the severe convective vertical shear present in developing deep moist convective atmosphere. As soon as I was advised that my route would take me through; rather than over; an area of developing thunderstorm clouds I should have rejected the clearance and negotiated a different route as I ultimately did. I also believe that more ARTCC personnel should be assigned to work busy airspaces when unusually severe weather (Hurricane was in area) is expected. That would make provide more time and flexibility to direct traffic safely through such areas.