we've done a lot of these in my classroom at school and we have access to a washer/dryer so my kids can wash them when they are done.
This is what I do at home and have them do the same in class.
We do not wash the quilt, we only put it through the rinse and spin cycle. We only dry one quilt at a time. I give my student a timer and have them set it for 10 minutes. We set the dryer on cotton, toss the quilt in, and start it up. When the timer goes off, they open the dryer, clean out the lint tray, set the timer for another 10 minutes, and start up the dryer again.
Since you are only drying one quilt, they actually dry pretty quickly. And when, after 10 minutes, there is little lint showing up in the lint tray, they can then leave the quilt in the dryer until it is completely dry. The whole process usually takes about 40 minutes, the last 10 being the final drying time needed. These aren't small baby quilts either, the kids make them to fit their size and they are high schoolers.