Being new to this mode of communication, I see that my response is late. However, more quilts are to be made. My question is, "how do you wish it to look?"
Do you want it to look as a pressed dress shirt? If so, don't wash it. But, keep in mind that all the dirt picked up during the process of making it remains in it.
What is wrong with puckering? All the natural materials used will shrink to one degree or another, and in varying amounts with each wash. Those of us old enough to recall wash days in years passed will remember having to pull fabric hard as it was ironed to make it lay flat. Hard to do when attempting to iron three layers. The puckering actually captures air which helps its insulatng factor, thus making it a warmer covering.
If you washed the quilt in the washing machine, that is a "no no". Quilt seams are very narrow. The agitating action of the machine can be strong enough to pull out the seams, thus a worse mess than puckering could ever be. Wash the quilt in the bath tub in which you use a very mild detergent in warm. not hot water. Using a gentle swishing motion, move the quilt to and fro, squeeze out, do not wring, as much water as much water as possible as the water goes down the drain. Rinse several times in either luke warm or cold water until no suds come forth. Squeeze, but do not wring as much water out of it as possible. Handle very carefully as wet fabric is very fragil. Put the quilt into the washing machine on the spin cycle. Once the spinning is finished, take the quilt out, lay it on a flat surface to dry. Do not dry it in the dryer as the heat will cause more shrinkage, thus more of the puckering which made you so ashamed.
The pictures you showed of what you considered to be a disaster leads me to believe that you are far too critical. The quilt looks very charming, and well done. Certainly, your mother-in-law should appreciate your thoughtfulness, and the time you spent to make her a work of art which she can display, and use, very proudly.
An addendum: Keep away from polyester, even small amounts. I used to have a fabric store, back in the days when polyester was the greatest invention since bread slicing machines. Some people who came into the store only had to walk by the fabrics with polyester in them to get a physical reaction--hives! True, fabrics with polyester in them look nice, but 100% cotton looks so much better.
Dana