In fairness to the OP - he doesn't say that he wants his wife to give birth here - merely that they want to visit and that the baby means they have to rush.
I would suggest to the OP that most Stupid will not let someone fly beyond 36 weeks.
If you wife is due in September - she will hit that marker at second week August at the earliest and end August at the latest.
EVEN if the appeal tribunal was heard in the next couple of weeks, there would then be a delay in getting the visa issued, which would almost certainly overrun the first date, if not the second - making travel impossible.
And even if the visa was issued, then the return would either have to be scheduled before 36 weeks, or you'd have to be prepared to open your wallet for the birth.
An example of the cost:
http://www.walletpop.co.uk/2011/01/17/b ... g-you-off/
This is 9 years ago - and the cost of the birth was nearly £5,000. You can expect it to have risen considerably since then. Any unpaid bill could easily result in further refusals.
So - to be honest - I'd suggest it would be sensible to forget the current visa quest prior to birth, focus on getting a healthy, safe, baby in Turkey and you head there for it.