The first thing we ATCO's know about TCAS advisories and resolutions is when the pilot tells us. At our unit, the controller will be alerted to an impending loss of separation by the STCA (short term conflict alert) but this is a totally separate system from TCAS. It is possible that they may become linked at some time in the future but they certainly are not at the moment.
(This Mexican airprox appears to have taken place on approach and the aircraft may have been controlled by the tower visually so there would be no STCA, even if it is available in Mexican ATC.)
Once a controller becomes aware of an impending loss of separation, he should give avoiding action based on what he sees on his display. As the display lags behind 'real time' by a number of seconds, the pilot may well be taking his own separation based on his TCAS warning which could be exactly opposite to what the controller advises.
At my latest safety training, we were advised that in the light of the Swiss mid-air, avoiding action is best given in plan rather than vertically. In effect, we give a turn, TCAS gives climb/descent. Of course, every incident is different and it can be hard to generalise but we are, in the UK at least, aware that pilots will obey the TCAS no matter what we say.