Having seen additional information given above some observations might be drawn from the plan view time-lapse sequence:
KLM 875, SE-bound at FL 330, did not change from his assigned FL.
BR 61, initially NW-bound at FL 320, did not change from his assigned FL but turned towards the West as instructed by ATC.
N8-840, initially SE-bound at FL310 whilst climbing turned towards the East as instructed by ATC.
It would appear that it was the turn instructions given by ATC that mitigated most effectively the risk of these encounters becoming too close.
As regards TCAS, it could be that the pilots of KLM 875 at FL 330 were advised not to descend, and that those in BR 61 were advised to neither climb nor descend - if indeed they saw more than a TA as at the closest point of approach they were diverging from the flight path of N8-840.
I would have expected the TCAS in N8-840, if operating, to have posted advice not to climb. As this aeroplane was relatively close to the two B777s I would have expected its TCAS display to have indicated the presence of the KLM 2,000 ft above and also of the BR 61 1,000 ft above on the reciprocal track.
I wrote, ‘if operating’ since the decision to initiate a climb and to have ascended 2,000 ft into close proximity of the KLM suggests that the display was either not visible or that the information it posted (proximate aircraft symbols and possibly TA/RA alerts) was not acted upon.
I do hope that more and accurate data is forthcoming!