Hawker 4000 First Officer reports speed and altitude deviations during Runway 24 departure from TEB with the Captain flying. At a speed above 250 KTS; the Captain begins descending with speed brakes until an aircraft terrain warning is activated.

Narrative

On departure off of Runway 24 in Teterboro; NJ; the Captain flying elected to use auto-throttles on takeoff. Our altitude alerter was set at 1;500 FT and the auto-throttle airspeed was set at 200 KTS in the manual mode. After takeoff; the Captain started a shallow climb and we accelerated rapidly.The Tower handed us off to the New York Departure Control. Then I heard the chime and saw the light indicating that auto-throttles had been disconnected. The Captain pulled the power back some at this time. As First Officer; I was listening to New York Departure and they were extremely busy. Because they were so busy; they contacted us and asked us to climb to 10;000 FT. I dialed 10;000 FT on altitude alerter. I looked at airspeed and we were over 250 KTS. I also noticed that the spoilers had been deployed causing the aircraft to pitch down. I told the Captain to climb to 10;000 FT multiple times.At this point; I looked down and noticed that the Captain was actually descending and that we had descended down to 1;100 FT. Departure Control contacted us and asked us if we were climbing. At the same time; the terrain warning system activated. I once again told the Captain to climb and at this point he finally started the climb and I replied to Departure Control that yes we were climbing. Then Departure asked us if everything was ok. I replied yes it was; at this time; since the Captain was climbing. For corrective actions; other than verbal commands; I was close to taking over control of the aircraft had the Captain not started the climb.