Currently we are migrating away from an AVST voicemail system to Exchange UM. With our current setup in AVST, we have call processors setup for certain individuals so they can dial 0 for the operator or 1 for the person's administrative assistant at anytime during the voicemail greeting. If no keys are pressed, it starts recording at the end of the greeting. Each person's call processor is a bit different because not everyone has the same administrative assistant.. Is there a way to do this with Exchange UM?

It may not be exactly like that, but quite close.  Ensure that the operator extension is configured at a dial plan level. Your users should be able to log into OWA and hit the little gear to get to their personal options.  From there, on the phone menu, you can set up call answering rules.  It will look like the below. There's no 0 option in the rules, but you can set that up in the dial plan, or on a per user basis in the user's UM settings from the ECP if you'd like.