Nil is an English word, which means zero, nothing. So that part is clear.
BaTTey or baTey (or baTe, or baTay) is a Hindi term from mathematics, which means 'divided by,' and is used most commonly as 'upon' part in fractions, separating the numerator and the denominator. The word actually comes from Hindi 'banTe' which means 'divided.'
SannaaTa is Hindi/Urdu for silence, mostly used to signify acute silence. However, the meaning of nil baTTey sannaTa has nothing to do with mathematics as such, because sannaTa, even when used as Zero, would give undefined result. Instead, nil baTTey sannaTa is a term to say 'nothing' in a proverbial, dramatic manner.
Though, living in Hindi heartland, I've heard Zero baTey sannaTa a lot more times than the rarely heard Nil baTTey SannaTa. Maybe it's part of the updated slang. And if not, I guess it could now be that.