Any prosecutor can claim the costs of the prosecution against a convicted defendant, including the Crown Prosecution Service. The CPS make a claim for costs in most cases but in reality where somebody is being sent to prison for a long time ie rape, murder, any application would be academic and so often is not made for practical reasons. Courts often make costs orders in favour of the prosecution where there is a non-custodial sentence or the defendant clearly has the means to pay.
The most common would be a motoring matter in the magistrates court where a guilty plea would costs somewhere between £50-100. Careless Driving guilty plea - fine a couple of hundred pounds, 3-5 penalty points on your licence plus costs of maybe £70 or £80 -v- negligent flying; fine £5,000 costs of circa £39,000.
Had the CPS prosecuted this helicopter pilot they would have asked for their costs and probably have got them. The difference is that they would have amounted to a couple, rather than tens, of thousands of pounds.
The CPS normally only apply for the costs of the actual proceedings in court. What they never do is claim for the costs of the investigation ie the costs of the police etc.