The DPP has to be satisfied that there is sufficient evidence to prosecute, otherwise its a waste of public time and money, and unfair to the defendant.
They can't make it ' beyond a reasonable doubt' because they don't know the defence argument in advance, but they do have to have a high degree of confidence in their evidence.
But as you've already said, circumstantial evidence is still evidence. Short of finding the murder weapon with finger-prints still on it and a video recording of the infant being killed then isn't it all circumstantial one way or the other. Even for something as relatively light as perverting the course of justice would be reliant on circumstantial evidence.
And the the DPP at the time saw fit to prosecute JH despite the fact that no matter what else happened she could not have killed Baby John. Today's DPP would have a much stronger case against Baby John's mother imho.
Any idea how long it usually takes for the DPP to decide whether to proceed or not, or could this go on for weeks/months/years yet?