Groovelottie - As has been suggested already, a lot is going to depend on whether you have any evidence to substantiate your claim that you did apply on time. With that, the LEA may accept that they have made an error and may give you the place you should have had, if they had processed your application. Or an appeal panel might allow your appeal for the same reason. But without any evidence that you applied on time, you are highly unlikely to win the appeal. It is inconvenient, obviously, to have two children in two schools, but the infant class size rule of 30 children per qualified teacher is strictly enforced.
Dobbs - As prh47bridge has said, your being adopted has no connection at all with school admissions. There is priority within the admissions code for adopted/looked after children but that does not extend to parents.
FrameyMcFrame - Essentially, in cases where the infant class size rules apply - ie where the admission number is a multiple of 30 and where there are 30 pupils per class in KS1 - you have to demonstrate some sort of mistake. That could be because the admission criteria do not comply with the admissions code, they haven't been properly applied, a mistake has been made which has deprived your child of a place, or the decision is so unreasonable that no reasonable person would have made the same decision and it can't be allowed to stand. Those are the only circumstances in which the panel is supposed to allow an appeal.
Where the ICS rules don't apply - where the admission number is not a multiple of 30, where classes in KS1 are less than 30 or in KS2 and secondary school - the appeal is decided differently, on the basis of the balance of "prejudice" (ie disadvantage) to the pupil in not being admitted and to the school in having to accommodate an additional pupil.