I worked at one before LO was born. We had a baby in my class who CD'ed. Obviously I was cool with it; it even influenced my decision to CD myself!
It didn't really change our diaper changing process much, because we are not allowed to re-use a cover. He had gDiapers and we just put the whole thing in a plastic bag and tied it off. He was required to provide his own closed container (they brought a diaper genie) and it was required to be emptied every day. If diapers were left overnight, the policy that his Mom signed stated that we could throw them away. She actually did forget once, but we didn't throw them away, lol. It was fine; she just got them in the morning. His Mom also taped a sign to the diaper genie that said in big letters, "CLOTH DIAPERS DO NOT THROW AWAY" so the janitors didn't accidentally throw them out.
I didn't mind at all, and neither did the other two teachers in my room. We were a pretty awesome drama-free team though. I REALLY can't say the same for every room, lol. I'm sure it varies by CDC as well too.
This is not really what you were asking, but after working at a CDC, I will not send my child to one. I have no idea how it compares to other centers that are not military-based, but I just know that it is not the environment I want for my infant. Four babies to one caregiver (even a wonderful caregiver, which most of them were) is too many, and especially the babies under 6 months cried a lot. It obviously varies by baby, and some do better than others, but it is just not a great environment for little babies.
When I was looking for a full time job, I decided I was going to go with a home day care where my LO would be the only infant and able to be held a lot more while the bigger kids were playing. I ended up finding a part-time job so DH is able to watch LO in the evenings while I'm at work, so it turned into a non-issue, but just thought I would give you my "insider" two cents. HTH!