I slept on LSD the first time that I ever did it. It took an hour or so, but I managed.. I don't remember having any interesting dreams, sadly.
As for mushrooms, I have only been able to fall asleep on them once, and that was after I smoked a fuckton of weed (which dissociated me and resulted in the STRANGEST trip) and laid down. I was sort of "dreaming" while awake, and eventually just fell asleep and these hallucinations carried over into my dreams. I can't even begin to explain how weird it was. But anyways, I suppose the weed made me a little tired, so I was able to fall asleep. Don't know if this may work for you or not but it did for me, and if you want to sleep on mushrooms then perhaps try giving it a shot.