PA-28 instructor reported an NMAC event during departure cruise with an inbound arrival aircraft on ATC vectors. ATC failed to provide traffic information due to heavy communication traffic.

Narrative

Departed ZZZ at approximately XA30 local; departed Runway XX; and was flying runway heading to 2000 MSL; after exiting class D turned to proceed directly to ZZZ1; after clearing the Bravo shelf above we climbed to 2;500 and were at cruise while proceeding northeast bound direct to ZZZ1; upon reaching the ZZZZZ intersection at XA45 local; without any warning from TCAS or ADS-B an aircraft what looked like a Cessna Skylane or Cessna 206 Came across our nose at the same altitude traveling west to east at approximately 50-100 ft. in front of us. We were the second to last departure to depart Runway XX; I believe the other aircraft was inbound to ZZZ at 2;500 to land Runway XY; and should've been communicating with ZZZ Approach and should've been vectored around us. The aircraft we nearly collided with was Aircraft Y; a Cessna 182. On frequency with ZZZ Approach XXX.X they had asked Aircraft Y to recycle and activate his mode-c transponder twice around XA30 local; the aircraft was then given a delay vector of 090 for the runway change at ZZZ; was then given a 310 heading; the ads-B record of the flight indicates the lack of ads-b and mode C capability of the suspected Cessna. The ZZZ Controller was quite busy at the time but never issued a traffic alert to the Cessna and most likely because he didn't know its position ultimately resulting in a near-fatal accident.