I am also sorry to hear the news. I hope that you have a reasonable alternative school.
I must correct some points that capers123 has made, because they do not apply to Bucks.
You may have had a really good case on paper, better than the others from the same school. On the day, though, other issues can be raised - extenuating circumstances - by the other parents that may have made their cases stronger than yours, and the supporting evidence from the school could have been stronger for the other children.
These points do not apply in Bucks non-qualification appeals - each case is assessed on its own merits, and you are not "competing" against other appellants in any way.
At the end of the day, there's a maximum physical capacity in the grammar schools, so some, often strong, appeals will sadly be unsuccessful.
Bucks panels do not take account of the number of school places available. If they were to allow every single appeal of the 900 or so heard each year, the resulting over-subscription would be sorted out via transfer (over-subscription) appeals after the allocations are made on March 1st.
If you've got strong new evidence, then you could ask for a new appeal.
A request from parents for a re-hearing will not normally be considered unless there is significant new evidence that was not - and could not have been
- available at the time of the first hearing.
As capers says, a procedural error would need to be sufficiently serious to raise questions about the panel's decision, and not be merely an administrative error that had no real bearing on the final outcome.