I came in on the very busy Valentine's Day without an appointment. My b/f had just proposed that morning and my nails were in need of an emergency treatment!
They said they could take me, but once I sat down I received a lecture about not having an appointment (by the way, they have a sign spinner outside. If that doesn't say "please walk in", I don't know what does). I apologized and explained the emergency. I didn't expect her to throw streamers or congratulate, but there was no acknowledgement. It wasn't a good enough excuse. I honestly would have rather they said they were booked. She made me feel so uncomfortable I thought about getting up.
The nails turned out fine. She was super rough and rushed, though. I mentioned how dry my hands were during the manicure, yet still had to ask for lotion once she walked away. She handed it to me to put on myself. If I knew how to do nails I'm sure she would have had me do the whole thing on my own.
TL/DR They'll still take your business when they are too busy, but will treat you like a total inconvenience.