One thing is sure. It falls within sapinda relationship and is prohibited under Hindu Law. But in some Southern States the marriage with uncle's son is prevailing and in some castes it is preferable. This is protected under the Hindu Marrige Act by virtue of custom. But, you clearly expressed that you both belong to North India, and as such custom does not prevail there, the marriage is illegal and void as per Hindu Marriage Act, 1955 and it cannot be registered under Hindu Marriage Act and Special Marriage Act.
But your problem can be solved very easily. As you both are unmarried, go to church and convert yourself to christianity and after that only marry as per chrisitan rites and customs. Your marriage will be registered and a certficate is issued by the authorized church, as such marriages are not prohibited under Christian religion. Now coming to the legality of such marriage, a magistrate in Delhi has fallen into the same predicament as you have been going through as love has smitten him through his cousin. But the Magistrate's father, Amrishpuri alike, and a known advocate was against this marriage and filed several criminal cases against his magistrate son and dragged him to several courts including the High Court of Delhi. Finally, the Hon'ble High Court of Delhi (judgment delivered by his Lordship Sh. Suresh Kait) approved the christrian marriage solemnized between the Magistrate and his cousin as legitimate one and put heavy financial fine with several strictures on the errant father of the magistrate for disturbing the peaceful matrimonial life of his son. Go to that judgment and get enlightened that how you can come out of your problem.
My best wishes.
