If the kids were born in UK, they are Brits......plus their dad is a royal Brit.
I’m not a lawyer, but would love to hear from one on this.
My impression is that if they are physically present in the US, then US law, and not British law, applies to them.
As an example, a Muslim who legally marries more than one wife in a Muslim country, and who then comes to the US, will not have his polygamous marriage recognized in the US, despite the laws of the country where it was performed.
https://www.bataraimmigrationlaw.com/polygamy-vs-bigamy-battle-cultural-bias-religious-prejudice