Is it permissible to use a paternal uncle's internet without his permission but with the permission of his son and by paying for the usage?

It is not permissible to use an internet service owned by another person except with their permission, based on the Prophet’s (peace be upon him) saying: "The property of a Muslim is not lawful except with his willing consent." Since the internet belongs to your uncle, you must seek his permission, and the permission of his son is not sufficient. Paying for the consumption does not justify violating the right without the owner's permission. Therefore, you must repent and seek forgiveness from the rightful owner.