Max rate as per lomapaseo, but obviously the acceleration can be slower if the thrust levers are move manually. Some aircraft / engines require a slower manual acceleration for takeoff in high crosswinds due to intake distortion which could adversely affect the surge margin.
Max rate / timings are determined in the air at approach speed. The thrust value used to determine initial GA performance is that achieved after 8sec; AFAIR max thrust has to be achieved within 13 sec.
Engine acceleration characteristics vary with engine design, and with perception of the primary thrust instrument; - N1 vs EPR, N2, N3, etc. Usually it is the engine core acceleration which is the critical control parameter.