C-172 Flight instructor reported near miss in traffic pattern as communicated by ATC.

Narrative

This Incident occurred during a go around on XXR initialed by my student; Flying the upwind; busy time on frequency; called go around; got no response; did not get any instruction from ATC either; turned crosswind still didn't get any instruction so we continued to fly standard traffic pattern and turned downwind. There was traffic inbound for right base for XXR and now we were on right downwind for XXR resulting in; what tower claimed to be a near mid air collision. As the frequency opened up tower immediately instructed i turn right and fly east i was advised of the deviation and instructed to land XXL. Contributing to this chain of events is due to a saturated frequency and increased airport activity at the time; may have caused ATC to miss my call or being occupied with other tasks overlooked giving me instruction on what actions to take as there where other aircrafts being issued instructions and traffic advisories at the time. In the future if no instructions are given i will continue my upwind as to not cause any inconvenience or near mid air collisions due to aircraft joining the traffic pattern for landing.