Just adding my thoughts.....I did the exact same thing as Meg's mom and it worked like a charm. I gave Frankie a little cat nap in the morning by planning an activity and then letting him fall asleep in the car for 20 mintues or so on the way home. It was enough to get him through to lunch and then after eating he had a second wind and could stay up until 1:00 to go down for a nice long nap. The first week, he took shorter naps and we had to suffer through that but then the key was we stayed consisitent and pretty soon he was taking a 2.5 to 3 hour nap.
As far as Meg needing a certain amount of sleep, I am sorry to tell you this, but that is going to change. They can't always sleep for 14 or 15 hours. The amount of sleep that they need is going to change, so something has got to give, either less sleep for a nap, or less sleep at night. Frankie takes nice naps, but he only sleep 10.5 hours at night on average. He just can't sleep more then that, and I don't think many kids can. So, big deal, we get up at 6:45. That's what happens when you have kids.
Try to look on the bright side, Meg is sleeping 12 hours at night! Do you know how many people would kill for that? It will all work out! Just stay consistent, but remember that along with that, you have to be flexible. Every day is going to be different for them at this age.
Good luck,
Kim