No, of course babies in nappies are not allowed in the swimming pool - it is unhygenic! There is a 45min session with Fisher Price on a few occasions during the cruise which parents attend with the baby. Babysitting is $8ph I believe.
You can bring buggy on board. Keep a few nappies in your hand luggage as it can take time for your luggages to be delivered to your room. Also when packing, split the clothes between the suitcases in case if one got lost, you still can have other clothes to wear.
Interesting response on the hygiene given that most people don't seem to bat an eyelid about very hot & sweaty (and sometimes very unclean looking) adults jumping into the pool next to them (without showering), which personally I find pretty unpleasant. Although kids are allowed into public swimming pools with swim nappies and it is the norm, they can't use the pools on the boat if they are still in nappies. I am may not particularly like it, however, we are booked on Voyager of the Seas in a few weeks and I accept the rule so my little boy won't be using the pool. In contrast Ocean Village had a paddling pool for chldren in nappies which was emptied and cleaned out everday. The Royal Caribbean Freedom Class ships also have a water area that children in nappies can use. Pearson's advice is good - one of our party's cases did not arrive until half way through the holiday on the last cruise. Have fun - we found cruising a very relaxing holiday with a small child.