As a nuc seller, I see a few options that can work for both of you.
But first I will note that it is not uncommon for nucs to startup emergency queen cells after the transfers into different spaces. The moves disrupt the pheromones of the colony and their first response can be to start cells. After a week in the new hive boxes, they stop doing so. Presence of cells after a nuc transfer does not mean there is no queen. Also, the queen may not start laying in her new home right away. She may take 1-3 days to get comfy and startup. Knowing elapsed time is important to determine if you are in this flux period or if there is indeed a problem. The condition you have could just be a sign of the disruptions. On new nucs I advise to destroy all cells in the first 2 weeks after transfer. Look closely though, especially if 5 days later there are still no eggs and queen cells appearing definitely means no queen.
Ok the swap options.
One is screen in the hive at night and bring the supplier the whole hive. Ask them to do the swap for you. As a novice, this helps prevent problems later if a virgin queen appears running around in there and wipes out your new queen. The nuc supplier should be experienced enough eliminate that possibility when he/she does the swap for you. This also gives the supplier some assurance as he/she will know exactly the condition and stability of the replacement colony when your return to pick it up. This is by far the best for you and for the supplier at this point.
If you have to re-nuc them for the exchange. Do this in the early morning at twilight, lighting just before sunrise. Take the nuc box over to the hive. Light up your smoker and give the hive a few puffs. Not much. Just a few drifts of smoke. Gently and quietly open the hive covers. Note which frames the bees are most clustered on. Transfer those frames, bees and all, into the nuc. Lift two frames at a time if you can, trying not to separate the frames to avoid breaking cluster as little as possible. Do not shake any bees. Just transfer clustered frames. Close up the hive and close in the nuc box. Whatever bees are left in the hive are yours for the inconvenience. Take the nuc straight to the supplier, or put it in a cool dark place until it is time to go. Store the nuc no more than 2 days.
Third option, if the supplier is close and schedules are flexible, perhaps he/she would bring the new nuc over and do the swap there for you.
Call, communicate, coordinate. It will work out smoothly.
Hope that helps!