Just realised that the BBC article contains part of a video interview with the parents (you can see the full interview on today's Victoria Derbyshire programme on iPlayer)... their whole focus is on this belief that 'UK immigration' is treating their adopted children differently than their biological child, but according to 'UK law' they should be treated the same.
But the thing is that they ARE treated the same... by both 'immigration law' and by 'UK law' (as far as I'm aware, there's only one law about this). All their children are entitled (or not entitled) to the same citizenship, unless of course, the biological son was born in the UK, making him a UK citizen that way.
Also, the wife has an English accent (only very slightly Americanised), so either the story is wrong and she WAS born in the UK and the kids are entitled to citizenship (but they'd have to register the adopted kids as British first), or she was born overseas but raised in the UK.
In the longer segment of the video, the husband says that the children were detained (i.e. implying the rest of the family weren't) and that they're either hoping their new visa applications are granted or that the government recognises that their mother is British and that they'll be given citizenship.
The wife also states that she cannot apply for visas for the kids to join her based on her being British (presumably because she cannot meet the financial requirement) and that the children are applying as dependants on her husband's visa.
Oh, and his visa is only valid until next March, and he says they're going to give him an 'indefinite' contact after that and so they'll need to renew the visas in a few months anyway ... though, actually, they could always switch to FLR(M) once he meets the financial requirement.