From the NZ MATS:
38.2 Traffic Advisory (TA)
If a pilot reports a TA the controoller shall advise the pilot of any measures that are already in place to ensure separation, if required, or reiterate the traffic information. If separation or traffic information has not been provided, the controller shall take immediate action to provide such.
38.3 Resolution Advisory (RA)
After being informed that an aircraft is responding to an RA, the controller shall pass essential traffic information or traffic information (as applicable) to the aircraft reporting the RA and to all other aircraft that may be affected by the manoeuvre. Control instructions shall not be issued to the aircraft responding to an RA or to any other aircraft involved with the RA event as this may interfere with the resolution action planned by the ACAS.
Once an aircraft has begun a manoeuvre in response to an RA, the controller is not responsible for providing separation between the aircraft that is responding to an RA and any other aircraft, airspace, terrain or obstructions.
I believe this is inline with the applicable ICAO standard/recommendation.
Basically, once an RA is reported, it's "hands off".
Bears a little thinking about, for in such a situation it is quite possible that the controller may have just become aware of a pending loss of separation, and be focused on preventing same (or worse).
Put yourself in the situation. You are rapidly communicating an avoiding action to one or both pilots, and extremely involved in the situation, and one of them says to you "TCAS climb" (or whatever the word is). At that point, watching the blips closing, you have to deliberately and immediately stop giving instructions, until it's reported as resolved.
The tendency to keep attempting your own resolution must be strong, for at least a second or three; the ability to instantly relinquish control is probably not well practiced, yet it is of high importance.
Having a notion in advance of such an event is, in my experience, more likely to make it easy to do the correct thing, when doing the correct thing runs counter to instinct.
(And just in case there are people reading new to this thread or the concepts within, controllers
are aware of what TCAS is for, and
absolutely do not consider it a personal safety net or backup plan.)