I'm going to tell my dad if he tells my mum something related to my mental health I'll cut him out of my life
What is going on with your mental health that you had to threaten your dad? Have you told him to not tell your mum about it?
I've told him multiple times already. He just doesn't get it!
Even if your dad is ignoring your wishes, it would be bad to cut him out of your life. If you cut him out of your life you will  both be sad and have regrets.
I don't care. He needs to understand I'm bloody serious.
Well, you need to tell him why you need to keep it a secret from your dad and if you are really loved by him, then he will respect your wishes. You should still tell your mother as well so that you have more people supporting you with your mental health issue. I hope you get better.