There may be others visiting this forum who will have some advice or previous experience that they can help you with. However, it is tricky to compare your case with anybody else’s.
You will need to find the exact basis that you son was not offered a place at this grammar school and appeal about that. Unfortunately, we have no ‘inside’ knowledge of Aylesbury Grammar School, apart from what is published on the internet. If you read their admission policy there is a clear over-subscription order for places and the last one, number 8, looks as if it applies to you – as you said you were not in catchment. School places would have been allocated using that policy.
I still strongly suggest that you enquire directly from the school about the prospects of a successful appeal and the process then take it from there.
All the best!