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I am a British Citizen by Naturalisation and my wife is an American. My first child was born in the United Kingdom when I was on a student visa, and my second child was born in America before I got Naturalised as a British Citizen. The two kids are with me in the United Kingdom.
Could you please advise if I can register both children as a British citizen?

UK born child - yes (entitled under Section 1(3)).

If
Wife and children have overseas born child have ILR and
Overseas born child over 13 have resided in the UK for at least two years, then
Yes (subject to discretion under Section 3(1)).

Thanks for the reply. My wife is not moving with us, hence the application will be solely depends on me as a single parent. Likewise, my abroad born child has no ILR and he is less than 13 years old. Any luck.

abbey20 wrote: ↑

Sun Dec 03, 2017 9:15 am

Thanks for the reply. My wife is not moving with us, hence the application will be solely depends on me as a single parent. Likewise, my abroad born child has no ILR and he is less than 13 years old. Any luck.

How did your child born abroad enter the UK? As a non-visa national entering as a visitor?

As a non-visa national entering as a visitor

abbey20 wrote: ↑

Sun Dec 03, 2017 9:15 am

Thanks for the reply. My wife is not moving with us, hence the application will be solely depends on me as a single parent. Likewise, my abroad born child has no ILR and he is less than 13 years old. Any luck.

It's likely that they may refuse the registration of your abroad born child.