You can do it in a day if you are arriving first thing. You would have to very organised, bring a packed lunch to save time, and you would probably be pretty tired though...
Waterloo is right next to all 3 attractions and the theatre is 15 minutes walk across the river. The atttractions have time slots, which which will help but you will still need to queue. I would call Merlin to ask about being fast tracked due to your DS's disability.
You can buy a combined ticket for all 3 attractions that is cheaper. Merlin attractions are also part of the 2 for 1 National Rail offer:
www.daysoutguide.co.uk/2for1-london
9am London Dungeon
I would start with the dungeon while you have energy, if that is what they want to do most of all. They recommend it takes 90 minutes to see the attraction so I would allow 2 hours. The queues should be short at that time.
11.30am London Eye
The ride is 30 minutes but the queue can take time (you can get fast track tickets though) so I would allow 1.5 hours.
1.30pm packed lunch by the river or walk along the river to the street food market behind the Southbank Centre (if the London Eye queues are short and you have time). There may well be free lunchtime music at the Southbank Centre.
2pm London Aquarium. They recommend allowing 2 hours so I would allow 3 for queues etc Or you could push it back to 2.30pm if you aren't that bothered about cutting the visit short and would rather have leeway to spend more time at the dungeon or for going to a restaurant.
5.45pm book at table for dinner at Joe Allen's (a theatreland classic, it's an American diner filled with signed photos and play posters) or somewhere on the Southbank/Covent Garden/The Strand. You can get an Uber to Joe Allen's or the theatre (if you decide to eat on the Southbank) for under a tenner. It may well be cheaper than the tube for such a short journey! It's only 15 minutes walk but an Uber might be less stressful and you might be exhausted by then!
7.30pm Theatre
At 13 and 15, could your DC cope with a post theatre dinner? When is the last train home? That would make for a more relaxing day. Joe Allen's is famous as a post theatre destination, especially for actors and theatre staff or you could grab fast food at Waterloo to eat on the train back. You could have a more substantial lunch in a restaurant with the extra time that frees up. If you do that, the London Eye is open until 6pm (the other attractions close at 5pm) so you could move that to last. But check the sunset time, if you are going later in the year!