Are you asking what the typical starting time is for any pump, or are you asking about cryogenic pumps? I ask because I have some experience with cryo pumps for LNG transfer. The problem is in the motor. Most have a severely limited rotor heating time meaning that if, on DOL statrting, they are not at full speed within 7 seconds the rotor overheats and swells, destroying the motor. Mind you that 7 second time limit was for the one that I worked on, it may be different for others. However, that was based upon the heating effects at DOL starting. We successfully used VFDs to extend the ramp time out to 15 seconds with no problems, because with a VFD the current is much much lower so the rotor does not heat up.
This isn't an issue on regular pump motors though. The acceleration time limits often have to do with pump lubrication via the fluid pressure created by the pump itself, especially verticals. If you are not at a certain speed fast enough, there is insufficient fluid pressure to lift the rotor or impeller off of the bearings, causing excessive wear. Every pump design that has this issue is going to have a different time requirement.
"Venditori de oleum-vipera non vigere excordis populi"