Based on the facts stated by you, the first issue is to determine what exactly happened when your cousin "got the Nikahnama registered." If a valid Nikah had already taken place in India in the presence of witnesses, Qazi, and other necessary formalities under Muslim personal law, then the marriage itself was already valid from the date of the Nikah. Registration is generally evidentiary in nature and serves as proof of the marriage; it does not create the marriage itself.
Therefore, if the marriage was genuinely performed and the Nikahnama accurately records the marriage that actually took place, the husband's objection that the registration was "against his will" may not by itself invalidate either the marriage or the registration. Particularly relevant is the fact that the parties subsequently lived together as husband and wife and have a child from the marriage. Such facts strongly support the existence of a valid marital relationship.
However, if the registration process involved any false declarations, forged signatures, impersonation, or procedural irregularities, then the husband may attempt to challenge the registration. Whether such a challenge succeeds will depend upon the facts and evidence.
The circumstances you have described may actually weaken the husband's position. If for years he represented himself as single in another country merely to obtain tax benefits or other advantages, that conduct would not ordinarily help him challenge a marriage that was otherwise validly performed. Courts generally look at the substance of the relationship rather than merely how one spouse chose to represent himself to foreign authorities.
The existence of the child is also an important factor. If the husband has acknowledged the child, lived with the wife, supported the family, or otherwise acted as a spouse, it becomes difficult to argue that no marriage existed.
If the husband files proceedings challenging the registration, the wife should preserve all evidence relating to the marriage, including:
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The original Nikahnama.
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Photographs and videos of the Nikah ceremony.
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Witness details.
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Communications between the parties.
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Birth certificate of the child.
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Passport, visa, school, medical, or other records showing the marital relationship.
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Any documents where the husband acknowledged her as his wife.
On the facts disclosed, a valid Nikah followed by cohabitation and the birth of a child is unlikely to be nullified merely because the husband later claims that registration occurred without his consent. The real question would be whether the marriage itself was validly solemnized and whether the registration accurately reflected that marriage.
Therefore, unless there was fraud or forgery in the registration process, the husband's threat that the registration is "wrong" does not automatically mean that the marriage or the registration will be set aside. The legal consequences will ultimately depend on the manner in which the registration was carried out and the evidence available regarding the original Nikah.