Actually just the opposite. O2 sensors have to heat up before they work. That is why your engine uses a default fuel trim when cold. It’s called open loop. When the engine heats up, it goes into closed loop. Many O2 sensors are heated so they begin to work sooner.
Edit: you paid a diagnostic fee. If they just read the code and made a guess, they ripped you off. You could have had that done for free at most car parts stores, i.e AutoZone. They should have looked up the diagnostic procedure for the code in the factory service manual or all data and followed all the steps. That will, in most cases, narrow down to the actual defect.
If you go to another shop, ask them exactly what they are going to do for their fee. You might also ask the mechanic you just used what he did beside hook up a code reader.