As a former teacher whose own son went through a tortuous and labyrinthine appeals process, only securing a place at a decent school a week into his first term and after he had started at the school we did not want, I can offer some advice.
The above poster who advised to change primary places for your second child so they can automatically attend the feeder secondary school attached to it is on to something. However, that won't work for the one you are writing about now.
What you can do, though, is get the older child's name on the waiting list for the school you want and keep it there. You would be amazed at how quickly these lists move. You can be at number one hundred and then be offered a place quite soon, mid term.
The reason is that once children are settled the parents do not want to move them. They listen to the children's pleas to stay put with their new friends.
Also, some people move away and others go down the private school route.
You need to be resolved and not to worry about a mid term move. At a good school this can be overcome by the end of the school year and anyway there are years to go before the GCSEs start.
Good luck and keep at it! Don't let the LEA think you are going to give up. When my son was given the 'wrong school' I phoned the schools' allocation officer and was charming and polite but kept asking when a place would be free at my desired school for my son. At the first term's week's end she said, 'Well, if you got in now with an application he might be in.' My husband took the day off work and the special letter for late application was on the doorstep when they opened up shop!