Thanks to @Majorjim! who showed this thread to me. Anyways, I have this exact same problem. So I built the N1-L3 rocket which has I believe 6-7 fairings and 4 of them have parts clipping through. It has almost 700 parts and weighs 1,000 tons (irrelevant). I click the green "launch" button at the top and the screen goes to the black loading screen with the little "loading" sign in the bottom right corner. This stays like that for almost exactly 20 minutes (yes, I actually timed that) and then it finally loads up on the launchpad. If the issue is really caused by the fairings, then I know 3-4 places where I can replace them with something similar which may reduce the loading time.
Munbro Kerman's post in Game hangs (Not Responding) on scene change to flight scene was marked as the answer