Sometimes they will page you to help wipe and all will be fine. Sometimes they will page you to help wipe and tell you your child isn't "fully potty trained" and can't be in the club. Almost happened to our tablemates, but it was the last evening, so they "let him stay" knowing he probably wouldn't need to poop again while in the clubs.
If club activity is important to you and your child will be three but isn't yet potty trained, you may want to consider postponing your trip until the "potty party" happens. Both my kids took over a year between "starting" to train and the point where I would have been able to get them into the clubs - and they were both 3 1/2 before it happened. You can lead a toddler to a potty, but you can't make him poop. As a result, my kids were 4 and 5 when we finally got on the ship - but I had no issues with kids clubs or the pools. Personally, and this is my opinion only, potty training was stressful enough without a deadline!
I can only speak to the club - as that was my kids age.
We registered them when we boarded (after lunch), although you can register them anytime. Your kids will be given a wristband and a nametage and you will be given a pager. You will need a password to move them out of the club, but anyone on the sheet with the password can do it - so if Grandma is going along, she can pick up. Your kids don't need to be with you when you register them, they can pick up their wrist bands later. The wrist bands will be on them (hopefully) for the whole cruise - if your kid is found wandering on deck 4, they will know who to call because they'll know by the wristband - so even if you don't plan on using the club, you might want to register and get a pager.
You will need to turn in the pager on the last evening, and there is a heavy fine if you lose it - don't drop it overboard.
If your child wants to leave, they should tell a counselor to page you. Sometimes things are a little busy and the counselors don't seem to hear the request (whether this is bad counseling or shy preschoolers, I don't know), so tell your preschooler to be persistant if they want you and you don't show in a few minutes. And plan on checking on them every few hours.
You'll need to be aware of the schedule - they don't check in or out kids during "movement" times - moving to and from dinner or sometimes the older kids go over to the lab. And you don't want to be the parent picking up your kids in the middle of a big event! So watch the kids schedules. But other than that, you can pick them up and drop them off anytime the club is open - and its open pretty much from 9am or something until midnight or something.
There are a few really special events. Our kids had their own sail away party - Lilo and Stitch came and taught them to hula. And Captain Hook taught them how to be a pirate and Snow White came to dance. They don't take pictures of these events, but parents often show with cameras.
They don't take the little ones swimming (the oldest lab set has a pool party - the 10-12 year olds?).
When they take the kids to dinner or lunch, they preplate some food - the stuff kids tend to like - mac n cheese, chicken fingers. People with strict dietary restrictions (allergies, kosher, vegetarians) or who really care that their kid ate something other than fruit cocktail and a cookie for lunch may not want to trust what the counselor is going to load on your child's plate, you may be better off making sure they eat when they are with you. Those of us who think a few days of nothing but mac n cheese and hot dogs and chocolate pudding won't kill our kids shouldn't have a problem.
You are welcome in the clubs yourself.
You can leave your kids in the club while you go on excursions - although you should tell the counselors if you won't respond to a page because you are off the ship or at Palo so they can set expectations.
They have a "show" the last afternoon. The kids are all on stage for about five minutes, Mickey comes out. They sing a song (M-I-C-K-E-Y M-O-U-S-E) and leave. They get a t-shirt. It takes all of half an hour for "practice" so set your expectations accordingly. But its very cute and most kids and parents love it.
In the evening, they pull out mats on the floor, everyone takes off their shoes, and its Disney movies until the wee hours. So you can pick up your sleeping child in the middle of Hercules at 11:00pm. If you see a little blond girl front and center - wide awake at 11:00 watching the movie, she's mine.