Pilot reported confusing ATC control instructions amid a heavy workload resulted in a NMAC.

Narrative

I was the pilot in command occupying the left seat and there was a second pilot in the right seat. After the right seat pilot approached the airport; entered the pattern and executed a touch & go on RWY 32L; I assumed control of the aircraft while climbing to make a full stop landing. The controller asked our intentions on climb out and directed us to make left traffic. After passing midfield the controller asked if we would be able to execute a short approach. We advised we would be able but received no response clearing us to execute the approach. After waiting approximately 10 seconds I contacted the tower and asked for and received a confirmation on the short approach; and began to turn a left base for 32L. While in the base turn we observed another aircraft on final approach for 32R but did not see this as an issue. At this point we both heard the controller direct us to 'turn East'. We did not receive a specific compass heading. I acknowledged the request and initiated a right turn to the East and started a slow climb. Approximately 10-20 sec later the controller inquired as to our position and I advised we were heading East towards Mt. Diablo. She acknowledged and advised us to watch for oncoming traffic. I continued a gradual climb and saw no aircraft in my immediate field of view. At that time my fellow pilot did see an oncoming aircraft slightly below us on a collision course. After being alerted I immediately increased our climb and banked right into what appeared to be clear airspace. The controller then contacted us and directed us back to join left downwind for 32L. At no time did the controller advise us that the turn to East off the left base was an error. The subsequent approach and landing included typical contact with the controller and were completed without incident or admonishment from the tower. During this period there were numerous aircraft in and outside the pattern and the controller was addressing what would definitely be considered an extremely heavy workload.