Well, I don't have a child "on the spectrum" so I cannot speak to whether his need for snuggling is related to that in any way. Perhaps it is, I don't know. It seems your husband doesn't think so, otherwise he would not relate your son's behavior to his own (unless your husband is also on the spectrum?).
But I am a little confused. I don't understand what reading time has to do with him needing you when you walk him back to his bed. Is he falling asleep while you are reading? If so, then you need to finish the story sooner and let him be groggy but go to sleep after you are out of his room.
As far as walking him back to his room, that is perfectly normal (kids not on the spectrum) and is what I'd recommend you continue to do. Don't pick him up or carry him, don't converse with him, keep it all low key and just "escort" him back to his room. Don't cuddle with him. Escort him back, maybe stand at the doorway for a moment (if you can do so without falling asleep standing up), and then go back to bed. He will stop eventually.
But if you carry him, talk to him, comfort him, snuggle with him, sit on the bed with him, etc... then you will prolong it. Because he will then be getting rewarded for getting up. You have to make it not worth it. So far so good, it sounds like.
I get that you'd rather your husband escort him back to his room, though, maybe? Instead of you having to get up. The only way I see that happening, is if your husband intercepts him before he gets to your room. If husband just talks to him from across the room and "tells" him to go back to bed, it won't amount to anything. And once he is in your room, giving it additional attention in any way (calling for husband to come get him, making a production out of refusing to escort him back to his room, etc) just strikes me as counter-productive.
Escort him back (or have Dad intercept him -- and escort him back using the same manner that you do) and eventually it should stop.
Good luck.
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ETA: Just realized, you also asked about Dad being able to tuck him in at night. Does Dad ever read to him? If not, why not? My kids always preferred that I read bedtime stories to them instead of their dad (maybe because he doesn't infuse them with as much "character" as I do, who knows)... but that doesn't mean that Dad never put them to bed. Granted, he did the entire bedtime routine differently than I did. And Dad wasn't always home at bedtime, so it largely fell to me, except on the odd occasion here and there, because it was just easier (for me) to just do it if I was home.
Maybe you could get Dad to put him to bed a few nights on the weekends (while you run out to the store or a movie with the girls or something). So son knows there is no alternative. When they get a "routine" of their own, then Dad can intercept him when he gets up later on.
That will help YOU by putting the onus on Dad. If Dad doesn't want you to schlepp him (escort him) back to bed, and wants you to have a firmer hand, then DAD can intervene by intercepting him and taking care of it. Otherwise, if he is unwilling to step up, then he needs to shut up.
My 2 cents.