I would imagine that there are either real concerns that the baby is not hers, or that she knew her baby would be removed or have heavy social services intervention in the UK and she went to Spain to escape this.
It's simply not true to say she wouldn't have gone to the media if she'd done anything wrong. Plenty of people do. They give statements and interviews and help the police, because they believe they will do better "hiding in plain sight".
It's possible that the hospital have made a big mistake. It's possible that she is lying about something, knowing that the hospital will be legally unable to reveal too many details.
If it turns out that the baby is not hers, though, or that she intends to put it at serious risk, the hospital would be hugely criticised for not investigating properly.
And there are phones in Spain. She should have called, even if the baby was already born by the time it arrived, to have everyone checked over. If her mum didn't have a phone, her neighbours would have.
It looks suspicious to avoid medical intervention during the birth and then present a baby that seems too old... She could have at least avoided that line of questioning if she'd got medical care during or immediately after the birth, as is normal.