If you don't have to transport the finished product anywhere, and the color of the diapers goes with your color scheme, you could just tie a ribbon or affix a strip of paper around the layer, but since I was covering each layer with tissue paper and having to pack the thing up in a box for transport, I just used Scotch tape to hold it together. Since you want the parents-to-be to be able to dismantle the cake and actually use the diapers, make sure the tape you use will come off without ripping the diapers.
Do the same thing for progressively smaller layers to create the size cake you want. (The pic below is the top layer before I curved and secured the diapers into the tighter spiral formation.)
Since I didn't want a green-and-blue color scheme (though that would be cute for a boy shower), I covered each layer with white tissue paper. It took several sheets in order to not be see-through, but I just laid them together and wrapped each layer like a present (a very oddly-shaped present, being essentially round and all).
Make the wrapping as smooth as possible (not very, if my experience is any indication) but it doesn't have to be perfect. You can also smooth down or hide the imperfections with your decorations.
I opted to go minimalist, since I had these nice ribbons that worked with the party's color scheme, but I've seen some that are decorated with strips of scrapbook paper and 3-D flowers, birds, or butterfly embellishments, and some decorated with small toys and supplies for baby--pacifiers, loveys, toiletries, etc.
It turned out alright, I thought, and I bet my next attempt will be even better! In any case, it works for me!
PS: does anyone else find the photo-uploader on Blogger's "updated" post editor extremely un-user-friendly? It's driving me nuts! Any way to make it easier, besides going back to the old version?