They were told they could not register the birth in St Lucia, as they could not prove where Louisa was born.
I don't see how Grenada would have the standing to issue any form of birth certificate to a non-Grenadian not born in Grenada.
It is possibly (though a PITA) to go through life without a birth certificate (some countries do not insist on registration, or it's patchily administered, or records are destroyed (eg in war/insurrection) or people have fled without all their documents and there's no realistic prospect of re-obtaining them)
So there are work arounds, but they may take time.
It looks like the consulate in Grenada have activated a work round by advising that a DNA result proving Louisa is the child of these British parents will mean they can establish British citizenship by descent for the baby, and get the documentation enabling her to travel to UK.
As the article says, the current hold up is the need to await the DNA results.