Hi Natalia,
Thanks for visiting the Disney Parks Moms Panel with your question. I love a stay at
Disney's Old Key West Resort. The vibe there is relaxing and laid-back, and the villas are spacious and comfortable.
You are correct in that check out time is 11:00 a.m. I did some checking into your question, and when I spoke to a cast member at Disney Vacation Club member services, I was informed that officially that there is no availability for late checkouts at the Disney Vacation Club resorts. However, I think that sometimes given circumstances and cast member discretion, a late check out might be granted. It cannot hurt to explain your situation and ask the cast members at Disney's Old West Resort and see what they say, as long as you are ready for the answer to be "no." If a late checkout is available, a cast member at Disney's Old Key West Resort told me that they would be complimentary up to 1:00 p.m., and that after that time may be available for a cost.
I remember traveling with a little one who needed to nap, and I know that logistically it can be tricky, especially on travel days. There are a few tricks for getting you child to nap at Walt Disney World even if you are not in a villa. Will your son fall asleep in a stroller? You could take a walk around the grounds of the resort to see if he will nap. You could also try taking the boat ride from Disney's Old Key West Resort to Downtown Disney - sometimes a peaceful float along the river and the calming motion will help a little one to sleep. Along the same lines, a ride on the monorail was known to lull my daughter into an afternoon nap.
I would like to thank my fellow Disney Parks Moms Panelists Andrea, Kari, Juliette, and Erin for their thoughts on this answer. Please do not hesitate to ask if you have any further questions, Natalia.