It would not change the ATC system at all, since TCAS and ATC work on completely different levels and have vastly different purposes.
We even have an ICAO recommendation clearly stating that ATC procedures should not be changed based on ACAS implementation:
The planning of air traffic services and the determination of the level needed for these services should not be influenced by considerations relevant to the deployment of ACAS in a specific airspace.
ATS Planning Manual, Doc 9426
(Note: ACAS is an umbrella term, covering different types of airborne collision avoidance systems, the most common of which is TCAS)
Think of the ATC system as the road network, traffic lights, road signs, and police. TCAS is a standalone device in your car that warns you the guy coming at right angles to the intersection in front of you is going to run the light and T-bone you. Whether you have such a device or not, it doesn't change how the traffic control system work.
TCAS has been around for 40 years and it didn't change the way traffic is controlled because it's a stand alone system that only impacts the crew on board the airplane that gets a TCAS warning.
TCAS is mostly to protect big aircraft from incursions from small aircraft, that are hard to see and that ATC can't always prevent. What got it going in the first place was the famous crash in LA where the DC9 on an approach overran a C-172 back in the late 70s.