Does he have some kind of therapist that he sees for his severe cognitive and speech delays? Or a doctor? I would present all of this to that therapist or doctor and ask for advice. The therapist may say "I can't say yes or no" (because sometimes they are reluctant to give advice, out of fear they will be held responsible if things don't work), but you could try reassuring the therapist that you are just sincerely trying to find out if this schedule is too much for your son -- especially considering that he has these delays.
Even if he does not need a nap any more, doesn't he still need "down time" or "quiet time" to rest and regroup and just be quiet during the day? The world can be a stressful place for a child with delays and he might need down time each day.
Would adjusting to two different routines be hard for him with his cognitive issues? Each school will have its own way of structuring the time; its own way of getting students from activity A to activity B; its own rules and consequences for breaking them; and so on. Learning just one preschool's routines can be tough for some toddlers, but learning the routines at two different places may be a real challenge and might prove confusing.
Personally I can't see sending any child to two preschools, unless it was absolutely needed as, basically, day care coverage for a working parent situation. For a child with his delays, I coudn't see doing it at all. Definitely talk to his doctor or whoever diagnosed his delays. Would he be better off spending time in therapy rather than in a second preschool?
Also you should check -- Head Start may not accept him if he is also enrolled in another preschool. I don't know all the rules, but Head Start is a government program aimed at lower income families, and the rules may say that kids whose families can afford another program cannot be eligible for Head Start as well. Check first before you enroll him in either program.