I think it's very similar to the way the term is used in the context of learning. In education "processing" refers to how kids come to an understanding of different concepts. It's more often used if there is a problem - as in processing disorders. Someone with an auditory processing disorder for example, does not process information just by listening - they need visual cues as well.

It's similar with emotions - if someone hasn't processed something yet then I understand that to mean they haven't come to a clear understanding/ acceptance of an event and how that event has effected them (among other things). People process emotions differently as well - some have a very fine tuned and rapid emotional response to events. Others have delayed reactions and may not emotionally respond until well after something has happened, if at all. I don't think a therapist has any way of knowing for sure if someone has really processed something or not. You can guess based on a client's behavior but I don't think they can ever know 100%.