Nick sounds like an immense arse and it looks like he's been manipulating situations so that Richard gets left out. And now he's totally got his own way because he's now in charge of driving and makes sure Richard is not included.
What about your part? You go along with going to the hospital with Nick, and don't tell Richard you're going. Because only two guests are allowed at one time is a bit of a flimsy excuse when you could do swapovers during visiting hours to make sure all three of you got to visit Dennis.
As others have said, Richard obviously has issues with social situations and how to behave. You and Nick could have arranged to pick him up and take him to the hospital and someone could have explained that this was the right thing to do and that Dennis would appreciate it. Instead of which, you've all left Richard to his own devices and when he hasn't acted in the correct way, it's been used against him.
When Nick decides to secretly organise trips out with only the three of you, you also went along with it. At least you felt guilty about it and have been seeing Richard yourself.
If I were Dennis, I'd be hurt and upset about Richard not visiting too, but can he not appreciate Richard's limitations? And perhaps because you and Nick were visiting Dennis but not telling Richard about it then perhaps Richard thought Dennis wasn't seriously ill, since nobody was visiting him.
I feel extremely sorry for Richard. He obviously struggles a great deal and thought he had friends in you and Dennis with whom he could share a hobby. Then Nick comes along, decides that he doesn't want Richard around and does his best to exclude him when he has the opportunity.
No wonder Richard seems obsessed with Nick driving him to events, since it's Nick's doing that this has all happened. Easy to persuade Dennis who had a difficult but tolerable friend that this friend is a waste of time and he shouldn't bother with him when Nick has manipulated things so well! Add on the hospital stuff and voila! Nick gets his own way completely.
I feel immensely sorry for Richard. He's obviously got a disability but he can no more help the way he is than a blind person could help walking into things, falling over things, and generally being really annoying if all their friends did was watch them stumble about, did nothing then start disagreements with them because they weren't walking in the right direction. But the blind person would have been helped from an early age by a stick, a guide dog, and other aids etc. Richard has been stumbling about for over 30 years when it seems clear he has some kind of disability.
Poor bloke! Annoying and frustrating as he is, and perhaps a friendship best kept to smaller doses, I feel for him.
If you want to try and mend some bridges then why not speak to Dennis and have a chat about possible reasons Richard might not have visited (including him not knowing about you and Nick visiting). I think it's a poor show to say well, he lives in the same town, why couldn't have gone himself. If you speak with Dennis and perhaps talk about possible reasons Richard might be as he is, and maybe both of you (without diagnosing Richard!) could be open to being a bit more understanding when Richard behaves in a way that isn't socially acceptable.
Then if it were me, I'd start organising things with Dennis and Richard and leave Nick out, then see what his reaction is when he finds out. I doubt it'd be pretty. I agree with the other posters who say Nick is a Wendy. He's a nasty piece of work imo.