The CPS had no say, Dickens. I wrote about it in my reply to Callistemon which refers to their position on the CPS. It was definitely a private prosecution by the RSPCA.
The RSPCA is an organisation. We know, from recent events, that large organisations can put reputation ahead of what is right. We don't know if that was what happened in this case, but they were getting a great deal of pressure from the media, both online and in print. It should, therefore, be considered. The RSPCA did not visit this horse at all. What, therefore, made them decide to pursue this woman? We can only guess what the CPS would do if given the information. They may have decided not to bring charges, or they may have decided to prosecute. We simply don't know. The RSPCA has had to apologize in the past for prosecuting to protect their reputation.
Add to this the presence of a group whose very name tells they are prepared to damage or destroy to prevent other success. We don’t know what their part was.
I am on no one's side in this, but I think we always have to be careful before jumping to conclusions that can/may ruin someone else's life.