What is the THS deflection speed? Let's restrict this question to best sellers airliners (A320 and B737). On  this aircrafts, the THS is used for pitch trim. The speed requirements may not be has intense as for elevator as it control a longer term action. I excpect the elevator to act quickly (I imagine less than half a second from full up to full down, several degrees per seconds) but what about the THS? Does this speed change depending on the flight phase? (I see no reason, but I may be mistaken)
This question is restricted to autotrim function (otherwise, I think the speed of pilots hands may be the limiting factor)
For the A320:


Conf 0/1 the rate is 0.3°/s
Conf 1+F/2/3/Full the rate is 0.7°/s


I haven't found any difference for A/P vs Manual flight.

In A319, THS freezes in the following situations:


The pilot enters a manual trim order.
The radio altitude is below 50 ft (100 ft with autopilot engaged).
The load factor goes below 0.5 g.
The aircraft is under high-speed or high-Mach protection.


In A320, THS freezes in the following situations:


The pilot enters a manual trim order.
The radio altitude is below 50 ft (100 ft with autopilot engaged).
The load factor goes below 0.5 g.


There are further conditions under which THS range is limited, let me know if you would like me to expand.

Source: A320 FCOM DSC-27-20