The deadline was for first round allocations. Now that has passed and everyone who applied has been allocated something, the rest of the remaining spaces in schools which have them become available for people to apply for, whether this is because they're not happy with what they have been given or because they've moved into the area since the original deadline (or simply because they missed the deadline... believe it or not this happens to people each year).
No object has been defeated as some schools are no longer available as they have no spaces left, due to being oversubscribed in the first round of allocations, hence the need to join a waiting list if you want them. Someone applying now isn't going to bump someone who already has a place out (but they may bump someone on a waiting list for that school down as the waiting lists are not first come first served). This is the fairest system as it allows priority to anyone who moves really close to a school after the original deadline. They may not get in to their local school straight away after moving, if it's oversubscribed, but they'll get offered a place if it comes up ahead of someone who lives the opposite side of town who applied in the first round of allocations and didn't get in. Basically, if you apply late but would have got in based on the schools admissions criteria had you applied first time round, you'll go straight near the top of the waiting list as a priority.
The only way spaces which have already been allocated in the first round become available is is someone rejects it. If a school is oversubscribed it is very rare for someone to reject it because they're not happy, as it's obviously a popular sought after school and they must have applied to have been given a space in the first place. However, people do move away and no longer require the place, or opt for private school instead. Some people's circumstances change, as hour's has. Places at oversubscribed schools do sometimes become available, so it's worth going on a waiting list but I wouldn't pin your hopes on this.
Does that make sense?
You won't win an appeal based on the school you wanted being closer than your allocated one, and causing a logistical problem for you now your other child is due to start nursery there. That's not their problem. Your first choice was just move popular (or has less spaces to offer) than your allocated school, hence your child getting a place at the other school despite your preferred school being closer. More children applied than they had places to offer, and other children who applied met the admissions criteria ahead of your child, that's all. The only way you will win an appeal is if you can prove the admissions criteria was applied wrong and someone got a space ahead of you who shouldn't have, which is very rare.
If a space at your first preference doesn't become available, does the allocated school have a nursery you could try to apply to for your other child, to make life easier? Or at least a nursery or pre-school nearby so you're not going in two different directions?