The DPP can intervene to prevent a private prosecution from proceeding under some circumstances.
www.cps.gov.uk/legal-guidance/private-prosecutions
In a State prosecution, the defendant who challenges the sufficiency of the evidence has one option: to apply to the Court to dismiss the case. In a private prosecution, the defendant has an additional option: to invoke section 6(2) of the Prosecution of Offences Act 1985, which entitles the Director of Public Prosecutions (“the DPP”) to take over a private prosecution so as to discontinue it. The DPP’s policy on private prosecutions states that she will discontinue a private prosecution if the evidential test from the CPS Code for Crown Prosecutors (“the CPS Code”) is not satisfied, i.e. there is no realistic prospect that the defendant would be convicted. The lawfulness of the DPP’s policy was affirmed by the Supreme Court in R v Gujra.[1] Their Lordships regarded the DPP’s power to discontinue a private prosecution on evidential grounds as an important safeguard against abuse of the right to prosecute.
On the matter of costs (from above link):
Earlier this year, Peter Gray, the former partner of Gibson Dunn & Crutcher LPP, received a substantial costs award after Hammersmith Magistrates Court dismissed a private prosecution brought against him and others concerning allegations of human trafficking. District Judge Tan Ikram reportedly described the private prosecution as “wholly improper”. The Judge’s rebuke is a reminder that private prosecutions, which have swung into fashion in recent times, need to be carefully handled.
www.corkerbinning.com/defending-a-private-prosecution/