It is a recognised point of law that when you make a booking in the UK that you are also accepting that company or business Terms and Conditions. You should have read the smallprint although a lot of people don't bother to do so. Alton Towers Terms and Conditions are laid out clearly in the link below
http://www.alton-towers-breaks.com/terms…
It quite clearly states that you will get a booking confirmation so when you didn't get one within 7 days you should have contacted them to find out why not. They even give you the option to change your booking or cancel without incurring any charges. They have no way of knowing if you have received your booking so it is up to you to say so.
When you knew your sister had received her confirmation why didn't you then get in touch with Alton Towers?
If you had paid by credit card you could have claimed from the credit card insurance but without a receipt even that is doubtful.
You can try the Citizens Advice but they will probably quote what I have written above.
I really, really wish you could get your booking cancelled or the booking moved but don't think it is going to happen. You are now relying on Alton Towers goodwill and they don't appear to have any.