Radio altitude is the primary ground sensor for the TCAS, and it's also a primary sensor for the GPWS. TCAS is not connected directly to GPWS, other than that, and prioritizing of audio alerts and warnings.
TCAS interrogates all transponders and military IFF within about 60 miles, and calculates distance, reported altitude difference and rate of closure in seconds. 35 seconds to CPA, closest point of approach triggers TA. 20 seconds triggers RA. Individual speeds of the planes doesn't matter, just their rate of closure. Angle of approach is irrelevant, also.
Sounds like the TCAS performed as it should have.
GB