If he was born in the UK, his parents were married at the time, and his father is British, he has always been a U.K. citizen.
Your children are also already UK citizens as well. They do not need to register as British or get British birth certificates... they are simply British already because your husband was born in the UK and just need to apply for UK passports.
So:
- your husband is a British citizen otherwise than by descent (because he was born in the UK), and so has already automatically passed his UK citizenship down to all of your children
- your children are and always have been UK citizens by descent because they were born in the US to a UK citizen otherwise than by descent.
(Note that if your husband had been born in the US himself, he would be British by descent and would NOT have been able to pass his UK citizenship to your children as UK citizenship can only be passed down to one generation born abroad).
So, in regards to citizenship and passports, all of them are dual-UK/US citizens and all you need to do is apply for a British passport for your husband and British passports for all your children.
However, the important thing is what is YOUR nationality? Are you British or American?
If you are British and have a U.K. passport, then in order to move to the U.K. as a family, all you need to do is apply for UK passports for all of you.
However, if you are American and do not have UK citizenship, then you will need to qualify for a U.K. spousal visa first before you can move to the UK.
The spousal visa has financial, accommodation and relationship requirements to meet.
For the financial requirement, unless you have substantial cash savings (more than £62,500), in order to qualify for a spousal visa, your husband will either need to have been working in the UK and earning £18,600 or more for at least 6 months before you can apply, or he must have earned at least £18,600 in the US in the last 12 months AND have a guaranteed UK job offer paying at least £18,600 and starting within 3 months.
For the accommodation requirement, you need to show evidence of guaranteed accommodation in the UK for you all to live together.
For the relationship requirement you have to show you are in a genuine relationship... I.e. are married and have been living together in the US
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