I get the routine thing, I totally do. I've done multiple trips with little kids, and there is one thing common no matter what their personality - Disney TIRES them out!
My little guy, who was on a firm schedule of one loooooong afternoon nap (which at home we HAD to keep) at 13 months became a cat napper at Disney. He'd conk out on a ride (um, Safari!) and wake up at the end. We'd be running from one attraction to another while he was in the stroller and suddenly he was sleeping again. Bus ride? Insta-nap. But then when we made the trip back to the resort to sleep...he didnt want to sleep. It was frustrating at first because we kept trying to stick to the afternoon nap because we thought we had to, but once we just rolled with the cat naps it was a lot easier. (And we got some afternoon pool time instead of sleep!)
My daughter, who at 3 stopped napping except for car ride type naps, upon our visit when she was 6 was super happy to go back to the room midday and sleep for a good hour or two every day. (We were touring pretty hard that trip)
The play it by ear comments above are all likely people who've run into the same thing - the kiddo and sleep thing is going to change a LOT at Disney, and you have to just roll with it.
Both had stopped using strollers but we still get them at Disney and only Disney - we rent a City Mini from one of the rental places, because it is very cushy, comfy, has a huge sunshade, and has a beautiful recline for long naps. Given that you're only there for a few days ... I'd probably try getting one and aiming for a stroller nap or two, and letting them nap when they want. I would just set your touring plans to include "downtime"... a loooong walk from one side of the park to the other after lunch usually does the trick.

Then plan to swap for big rides or something, or grab a snack, etc.