I can't help with a 'how long from show to labour' guess I'm afraid - but I've got an ELCS booked for my 1st, and was told firmly by the head MW what to do if I thought I was going into labour beforehand: -
If you are in two minds as to whether or not you are in labour, get to hospital and don't wait. If you are in labour and post 37 weeks, as you are, they will still do CS, just obviously earlier than planned!
If it turns out you aren't in labour, fine, you can go home and relax again.
If you ring the hospital and speak to a MW, and they tell you to stay at home/see how it goes/have a nice bath etc, ignore them and go in. This would be perfectly good advice for women aiming for VBs in early stages of labour, who are generally advised to stay at home as long as possible, but not if you have a planned CS. Things could advance labour-wise quickly, and you don't want to be turning up 7 cm dilated or worse. [unless you are happy to have a 'trial of labour', which I'm not, and it doesn't sound as if you are either].
Take your notes (obvious, sorry), and if you have an ELCS booked and agreed, and that's what you are happy with, don't get drawn into 'why don't you see how it goes' conversations. Be firm and refer them to your notes.
That was the advice I was given. I know it doesn't quite answer the question you were asking, sorry, but it's in the same 'what if' scenario. Good luck.