We were OK this time as had a sibling at school, but went through this 5 yrs ago. A couple of things worth remembering:
The schools you didn't get into should be able to tell you where you are on waiting list by end of week
Also, don't assume that if you are at no 5 for example, 4 other kids will start before your child. In about 2 wks time when everyone has to accept places, everything will change again. Some of the kids may not take up offers, then those at top of waiting list may actually turn their offer down (they may have got 2nd choice and be ok with that!).
Lastly, LEAs do make mistakes - when ds started, 2 of the children got offers from more than one school in the same LEA - although rarer than above, it does happen.
Best bet is to find out where you are on waiting lists & take it from there. If you are in top 5 there is a reasonable chance you could still get a place by Sept. If not soon after.
With ds, we were no 2 on waiting list for our 1st choice school & got an offer when all the paperwork went back in the May. Another friend was no 9 though and had to send her child to another school for a year before they got an offer.
From experience, appeals don't yield fantastic results but if its on the grounds of medical issues then it's worth getting stuck in and going for it. If its on distance/catchment alone then it doesn't look so good.
Sorry, if its not what anyone wanted to hear, but this is just my experience from the whole process. I'm in SW London where pretty much every school is over subscribed.