pilots continue to fly usually the previously assigned route. There are a bunch of rules designed to minimize the risk of collision.
When there is visibility, old fashioned see-and-avoid is very much in play. There are basic right-of-way style rules that prevent collisions. However, even in good visibility it is hard to see other planes at typical distances and speeds.
Most planes, and all of the commercial passenger liners, have TCAS based on ADS-B. This is a peer to peer transponder system that can issue resolution advisories with guidance to both aircraft to prevent a collision. No ATC infrastructure is involved.
ATC towers have red/green light guns that they can use to signal pilots. There are defined signals to issue instructions and landing clearances.
The 3D volume of airspace is so vast that collisions are normally pretty difficult. En route traffic is channeled onto airways, but the airway corridors are usually at least 5 NM wide, and traffic in opposite directions are on alternating altitude assignments. The main exception where collisions are risky is in the terminal area, where a whole bunch of planes are trying to get to the same limited runways. That's where collisions happen and where ATC is the most important.