At least in FS rotation was around 180 knts and takeoff was generally 220 knts. If you take 2/3rds of a 3 km runway thats 2000 m = 1/2AT^2 : V = 105 m/s thats a minimum average 0f 2.8 m/s^2. At liftoff you hit the afterburners on a full load, that would carry you off the ground.
What's worse is that a concorde needed to be all but full throttle to maintain altitude after liftoff without afterburners just to maintain altitude below 10,000 feet because of the IAS limit. The bird did not fly pretty below 250 IAS. It was an accident waiting to happen. The angle of attack of the aircraft was appreciable and the afterburners were providing the vertical thrust component that really got things off the ground, the wing shape being ideal for a ground effect also, but once in the air its balancing act to keep it up without afterburners.
I generally tried to manage at 245 IAS until I was close to my climbout point, then would punch up to about 9500 and do a reasonable nose down to bring the AS to about 400 and about 350 kts drop the burners and climb to about FL300, again hit the AB and burn to Mach 1.7 around FL450 and let is drift up to flight altitude (there's no much airtraffic about FL440 so . . . . ).