Hi Sara,
When our family visits Walt Disney World, we savor those midday naps! Our preschool princess always needs time to wind down midday. Our 9 year old prince doesn't nap, but appreciates the down time in our room to either rest or make plans of where he would like to go later in the day. Whether you decide to come back to your room for naps or resting time is entirely up to you, but our family has never regretted doing so. It's kept us from many meltdowns within the parks.
As far as getting through the theme parks, that depends on which park you are visiting. We've been able to get through some parks in 15 minutes or less, like Disney's Hollywood Studios, or up to 30 minutes on a busy day at Magic Kingdom.
If you decide to stay in the parks and have your 2 year old nap in the stroller, be sure to take advantage of the
Baby Care Centers within each park. These are quiet spots that provide amenities for parents with infants or toddlers. They have nursing rooms, feeding areas, changing rooms and even a small shop where you can buy baby items that you may have forgotten. As a mama, I just love that it's a place to catch your breath. You're welcome to stay as long as you like, so bringing your sleeping toddler inside for a while would be a perfect alternative to leaving the parks all together.
I hope this helps you, Sara. Feel free to visit us again soon with any other questions you may have. We're always here to help.
Have a fantastical Disney day!