> Why would I admit to something I didn't do? I was confused and scared
You've just answered your own question. Your confusion and fear, not to mention the fact that you didn't have legal representation, allowed the police to bully you into accepting a caution; an easy "result" for them to improve their crime statistics.
> What else could I do? I accepted the caution instead of going to court.
Again, you've answered your own question. You could have gone to court, where your lawyer would have called your mother as a witness and she would have no doubt testified that you didn't hit her. On the assumption that there were no other witnesses of the alleged assault, the case would almost certainly have been thrown out at that point. In fact, chances are it would have been dropped by the CPS before it even got to court...
> So what I'm being told is even if I didn't do it, that it's done now so it'll ruin my chances of doing the work I want to do.
Well it's hardly going to help, is it! In my opinion, your best (i.e. only) chance is to be *completely* honest on your application and in your interview, on the unlikely event that your application even gets as far as an interview...
I know that this is not what you want to hear...
Again, you *REALLY* need to seek *PROFESSIONAL* advice.