In such a scenario, the best a controller can hope to do is to pass traffic information and give vectors to separate the aircraft horizontally so that TCAS can step out
In the UK, a controller should not be doing the second of those actions, and the first only in specified circumstances.
MATS Part 1 refers:
GEN Chapter 9 Para 5.2
When a pilot reports an ACAS RA, controllers shall not attempt to modify the aircraft flight path until the pilot reports 'Clear of Conflict'.
Para 5.3
Controllers should not routinely pass traffic information to aircraft conducting RA manouevres, or other aircraft affected by such manouevres. Nevertheless, there may be circumstances where the passing of traffic information is justified; consequently, controllers may provide traffic information under the following circumstances;
a) to aircraft conducting an RA manouevre if it is considered essential to flight safety (e.g. information on aircraft which are known to be in close proximity that are not transponding Mode C information).
b) to other aircraft affected by an RA manouevre if judged necessary by the controller (e.g in airspace where the carraige and operation of TCAS and/or SSR transponders is not mabdatory).