I'm no great defender of the police but I have to admit that, IN THE FIRST INSTANCE, the police got it RIGHT.
As an analogy, if the parents of a very young child reported them missing, the police MUST immediately consider the possibility that the child had come to harm at the hands of one, or both, parents. (That's simply because it's happened in the past). So, while of course carrying out other enquiries, they MUST question the parents as possible crime suspects. It would be madness not to do so.
Therefore, when Gaia Pope went missing, the police were OBLIGED to consider anyone who'd been with her shortly before her disappearance as possible crime suspects.
However it's at that stage that things seemed to go wrong. If a police officer knows the identity and address of a possible suspect of a crime then he is NOT allowed to arrest them unless he's satisfied that they won't attend a police station voluntarily. [s24(4), Serious Crime and Security Act 2005].
So, unless the family members had refused requests to attend a police station to be interviewed, their arrests were UNLAWFUL.
It's not entirely clear though as to how their names came into the public domain. I've not seen any statement from the police which actually named them, so perhaps the press got their names from social media (or even through those family members making public statements to protest their innocence). In which case the police couldn't be held to blame for them being named in the media.