The question is not how long have they been without a queen, but how long have they been without brood. Yes, the colony has dwindled in size, but a well-mated laying queen should get the colony back on track provided the population isn't extremely low, which you are indicating that it isn't. The only real problem is if the workers start laying, because they won't accept a queen at that point. Open brood, however, will prevent the workers ovaries from developing. If you decide to just purchase a queen, I'd still ask your mentor for some open brood, since that will also give the population a boost, along with keeping the workers from laying. You may have to go in and remove queen cells, if they build any, before you introduce your purchased queen. The few extra days that the delayed release of the mated queen will take is neither here nor there.
What I might be inclined to do in your instance is combine the nuc with your full-sized colony. That will increase the population as well, and you'll only have to worry about getting one colony queenright. Once the situation is resolved and the colony is built up, then you could split them and be back to two colonies. Or maybe you'll catch another swarm or two.
I poured them into the new Nuc box and my 86 year old mentor slammed the lid down before I could stop him. There were piles of bees on top of the frames and I heard the crunch of bees. I was concerned immediately he could have crunched the Queen.
I'm sure this is probably clear to you, but just to reiterate what you are already thinking, there is never a reason to slam down any equipment when working with bees. All your movements should always be deliberate and controlled.