Snezzy is right. Your mum is probably having a hard time deciding what to do. On the one hand, she is obliged as a good parent to teach you responsibility and that there are consequences to your actions (or inaction). On the other hand, she probably really does want you to go to Pony Club camp. Since you are signed up, she obviously thinks that Pony Club, and the camp, are worthwhile activities for you to be involved in. Besides the fun parts, there are so many ways that Pony Club makes you into a better person.
First, you have to take complete responsibility for the things you forgot. Saying that you have a lot on your mind, or are really busy is not a good excuse. The only fact here is that you failed. It's your fault, no one else's, and not your circumstances. Don't try to make any excuses to justify why you didn't do these things, just apologize. As a parent, the last thing I'd want to hear is why it's not your fault. It is your fault, just come to terms with that.
Continue to try to be proactive the rest of the week and get things done before being asked. Try to prove that you are really responsible, despite your recent setback. If it comes up and they're still not sure what to do, offer to be punished in a different way for your failure, instead of being denied camp. Make sure it's something that is important to you, so you look like you're really taking it seriously.