Initially, they were told by a hospital they couldn't register her birth because she was more than 24 hours old.
It sounds like if they'd presented earlier at the hospital it would have been fine (excluding huge risks doing a birth with no medical care with them or nearby)
It does make you wonder why they didn't do basic research before hand.
I can see why a DNA test is needed - as British citizenship is inherited has been for years and as they have no witnesses to birth - they need some formal proof child is entitled to a UK passport. I can see being passed round different departments would be frustrating but given situation entirely predicable.
It does make you wonder wtf they were thinking or why they'd think rules didn't apply to them.