I'll just preface this with the disclaimer that I am not a lawyer , but do work as a forensic scientist and expert witness (not in field of child protection, though).
My understanding is that the government are not legally liable (morally liable, maybe, but that's a different story...) because they instructed an independent expert witness who was not a goverment employee.
If the police or prosecuting authorities want to instruct an expert in a field such as fingerprints or DNA analysis they would approach the FSS (Forensic Science Service) which is a government-funded institution. If an FSS expert cocked up on this scale, the government would presumably have direct legal responsibility.
But many specialised fields are not covered by the FSS, and in such cases the prosecution will instruct an independent expert in the same way as the defence would.
I don't know what the legal position is with regard to the responsibility of the instructing party in checking the expert's qualifications, but I know that there are complete charlatans out there who still get instructions, simply because there is far more demand for this kind of work than people qualified to carry it out. Sadly, if people become reluctant to do expert witness work, this situation is likely to get worse, not better, as reputable experts are under such pressure that the field is wide open to charlatans. Also it is difficult for anybody who is not a specialist to evaluate an expert's qualifications unless they work in that field themselves, so solicitors and police officers who need to instruct expert reports are in a rather invidious position.
As an independent expert myself I have Professional Indemnity Insurance against the eventuality of being sued, much as an architect would incase his or her buildings fell down, and all other experts I know have the same, either in their own right, or through their institution (if their main employer is eg. a university or hospital). Having said that, although I can see that being sued is a theoretical possiblity, I've never heard of it happening. I imagine it would be very unlikely to succeed, since an expert is the one person in a courtroom who is allowed to give an opinion, rather than confining himself to matters of fact. This means you'd probably have to prove negligence and bad faith of some kind, which would probably be quite hard.
Sorry if this sounds really dry and uncaring. FWIW, I am horrified by some of the things I have heard coming out of these cases, but sadly I suspect they are unique only in the awfulness of the ramifications for children and families.