What a terrible spot to put you in, all three of them! I'm not even allowing my mother in! She is terribly pushy and trying hard to put the guilt trip on me. I don't have a close relationship with my mom like a lot of ladies here seem too. My support is my husband and its our choice. My hubby is supper supportive, but I have birthed before too. No way in hell is my mom who I consider stressful and creepy going to set the ambiance in my babies birth!
If mommy is stressed so is the baby and it can cause a lot of unnecessary complications. Trust your instinct, I personally am way to private to even see a male doctor or most health professionals. I birth at home, with this baby we want to do it alone. I don't want anyone bothering our intimacy.
Besides, your going to be in a lot of strange positions, your going to expel stool, your going to sweat, grimace, cry, laugh, have uncertainty, bleed, possibly tear and what ever else. Animals don't want to be seen doing this, which is why they hide, so why would you. You could just say something like. "well if you want to be in there you have to be naked too and I'm not comfortable with seeing you naked nor am I comfortable with myself being naked in front of you." I love saying things like that which embarrass people back into place hehe.
Best of luck, its your birth, enjoy it.