There is no specific relation. They're just a distant relative. But if you are very close to them, you would often call them your cousin, even though that isn't the case.
your aunt but it depends if it is by marrige
aunty
Probably your niece-in-law.
Your fathers brothers are your uncles. You are their nephew, or niece.
your cousin
your uncles daughter would be your cousin
What is my godfathers niece to me
Your niece's brother is your nephew.
Your son and your niece are first cousins.
Nothing. Your niece in law. Your brother in laws niece.
cousins
your niece
Your Aunt's niece is you (if you are female), your sister, or your First Cousin.
Disowning YOUR uncles is not something that an unrelated person can do. Also while you, as their nephew or niece, can simply decide never to see or communicate with them again, there is no legal means of disowning them, they will still be YOUR uncles.