Well, it's less a question of products not playing well together and more a testament to the durability of some polymers. More on that in a minute......
What happens if you start using a paint cleaner or compound without stripping the wax first? It depends. If your last wax application was a carnauba applied 3 or 4 months ago, there probably isn't a whole of it left on the paint anyway so the answer is, not much. A very recent application of a carnauba product won't stand up to compounding for very long either, and it's very doubtful that it will literally gum up the works.
As to those durable polymers.....if you've recently applied a very durable polymer/synthetic sealant like Meguiar's Ultimate Wax, or Blackfire Wet Diamond, Menzerna Power Lock, Zaino, etc then those polymers can actually slow down the initial cutting ability of the compound, and to a surprising degree. Heck, if you've laid down one of these products very recently and then try to damp sand with 3000 grit on a DA sander, it will even slow down that process. But read that carefully - it will slow down the process initially. Oh, you'll still get through it fairly quickly, but if you look at the surface closely you'll be scratching your head for a few seconds at least. You should see how Jason Rose reacts if you threaten to spray some Ultimate Quik Wax onto a surface just before he does a damp sanding demo - it's kind of funny!
I have to say, though, what I find interesting about this thread is how many of you responded basically the same way, that the presence of a wax is inconsequential if you're going to compound, when so often we see people asking about the best way to strip wax before doing any correction on their car. We agree with the consensus here in that, for the most part, you really don't need to worry about it. After all, how many of us actually apply a wax or sealant and then three days later decide to do a full correction on the paint? In the vast majority of cases aren't you doing a full correction some time after applying whatever LSP you commonly use, meaning that only a small percentage of that LSP is all that really remains? And in many of those cases, aren't you claying, too? What time and the elements have left behind, clay is sure to finish off. And even if you don't clay at that point (because you deem in unnecessary) the abrasives in your compound will dispatch with the small amount still remaining rather quickly.
Just don't go spraying any durable polymers onto the paint immediately prior to correction and you're fine.