From what you've said, I agree with PRH
If I've understood your posts
30 places were offered on offer day.
At this point you were technically in spot 31 BUT the LA discovered they had made an error in another application, so you would have really been at spot 32 if this error had not been made.
The LA offered the place for the error, class of 31, you at 1st on waiting list
IF the LA had withdrawn the "extra" offer (which as PRH said they only have a couple of days to do this in the case of an offer made in error, there is a LGO ruling on this - it's not in the admissions code) then it would have been a class of 30 & you 2nd on waiting list.
So either way you would not have had a place on offer day, in spite of the LA's error.
What went on after that with the are-they-or-aren't-they late applications is neither here nor there as far as the appeal goes. If offers had not been made then it would have bumped you down the waiting list anyway.
I know it's far from ideal having children at different schools, but it's not something appeals panels can take into consideration & TBH that's the risk you run when you apply for out of catchment schools where in catchment children have priority over out of catchment siblings. You will need to find a solution for the school run (childminder etc) as neither school will be happy with a child arriving late or leaving early I'm afraid.