The LA can fight all they want. They won't win.
Seeker, as I've said (a few times) before, 'anyone below her' would not be in a position to appeal (although any of them could appeal because every parent has that right regardless of how spurious their reasons).
The OP's case would be dealt with by 3.2 of the Appeals Code* (see below) - the panel would have to conclude that although they have the right to withdraw under clause 2.12 of the Schools Admissions Code 2012 (2.12 Withdrawing an offer or a place ? An admission authority must not withdraw an offer unless it has been offered in error, a parent has not responded within a reasonable period of time, or it is established that the offer was obtained through a fraudulent or intentionally misleading application. Where the parent has not responded to the offer, the admission authority must give the parent a further opportunity to respond and explain that the offer may be withdrawn if they do not. Where an offer is withdrawn on the basis of misleading information, the application must be considered afresh, and a right of appeal offered if an offer is refused.), the appeal cases which dealt with that issue previously, deemed 3 days too long to wait before withdrawing. The OP's had over 5 days of legitimate expectation of a place.
The first person on the waiting list could also appeal, stating that if the LA hadn't made the mistake, then they would have been admitted instead of being on the waiting list. Clause 3.2 b) would be shown to be false. They did not apply the admission arrangements correctly. Therefore, that family would win under clause 3.5(a)"a) it finds that the admission arrangements did not comply with admissions law or had not been correctly and impartially applied, and the child would have been offered a place if the arrangements had complied or had been correctly and impartially applied."
The other 180 children could not argue that they would have been offered a place. Therefore they would not win the appeal on that basis, although any of them may be able to convince the panel that there is space for them and it would not prejudice the school to take them. If there were lots of them, then it would come down to who had the best case.
*The Appeals Code 2012 says:
"3.2 The panel must consider the following matters in relation to each child that is the subject of an appeal:
a) whether the admission arrangements (including the area?s co-ordinated admission arrangements) complied with the mandatory requirements of the School Admissions Code and Part 3 of the School Standards and Framework Act 1998; and
b) whether the admission arrangements were correctly and impartially applied in the case in question.
3.3 The panel must then decide whether the admission of additional children11 would prejudice the provision of efficient education or the efficient use of resources.
3.4 In all cases, the panel must refer to the local authority and the admission authority (if the appeal is for a school that is its own admission authority) any aspects of the admission arrangements that do not comply with admissions law.
3.5 The panel must uphold the appeal at the first stage where:
a) it finds that the admission arrangements did not comply with admissions law or had not been correctly and impartially applied, and the child would have been offered a place if the arrangements had complied or had been correctly and impartially applied; or
b) it finds that the admission of additional children would not prejudice the provision of efficient education or efficient use of resources.
3.6 However, in multiple appeals where a number of children would have been offered a place, and to admit that number would seriously prejudice the provision of efficient education or efficient use of resources, the panel must proceed to the second stage.
3.7 The panel must proceed to the second stage where:
a) it finds that the admission arrangements did comply with admissions law and that they were correctly and impartially applied to the child; or
b) it finds that the admission arrangements did not comply with admissions law or were not correctly and impartially applied but that, if they had complied and had been correctly and impartially applied, the child would not have been offered a place;
11 At this stage the characteristics and circumstances of the particular child in question will not, except in extreme cases, be relevant to the question of whether the admission will cause prejudice (R (on the application of M) v Haringey Independent Appeal Panel [2010] EWCA Civ 1103).and it finds that the admission of additional children would prejudice the provision of efficient education or efficient use of resources.
Second stage ? balancing the arguments
3.8 The panel must balance the prejudice to the school against the appellant?s case for the child to be admitted to the school. It must take into account the appellant?s reasons for expressing a preference for the school, including what that school can offer the child that the allocated or other schools cannot. If the panel considers that the appellant?s case outweighs the prejudice to the school it must uphold the appeal.
3.9 In multiple appeals, the panel must not compare the individual cases when deciding whether an appellant?s case outweighs the prejudice to the school. However, where the panel finds there are more cases which outweigh prejudice than the school can admit, it must then compare the cases and uphold those with the strongest case for admission. Where a certain number of children could be admitted without causing prejudice, the panel must uphold the appeals of at least that number of children."