So, have you actually moved house now?
When did you move?
Did you send the LEA evidence of your new address? When?
The advice from the LEA to put your mother's address on the application form is very odd - if you haven't actually been living with your mother, in most LEAs that would be regarded as fraud!
It is true, as you say, that most school places (apart from looked after children, social and medical need and siblings) are allocated according to distance to school from the parent's address. Parents' work and family situation aren't taken into account. The LEA can only use the information that they have - if you moved some time ago, it would have been better to have told the LEA immediately, because there might have been the option of making a late application from the new address.
You will have seen from the other threads that in infant class size appeals - which yours will be because there are 30 pupils in the class - you can win only if you show that the LEA made some kind of mistake. It doesn't sound as if there has been any mistake here unless they ignored evidence of your new address after you moved nearer the school (and even this is complicated because there are fixed dates after which changes of address can't be registered, so we would need to know more about when you moved and when (if at all) you notified the LEA).
At the appeal, you can try to persuade the panel that the decision to refuse you a place in this school is unreasonable - but unreasonable means so unreasonable that no other LEA would have made the same decision, so the general inconvenience of not having a place at this cool school won't be enough.
What should happen now is that, once you submit evidence of your new address close to the school, you should move up the waiting list, as it is held in distance order in line with the admissions criteria.