I refused induction with DD2 after an emcs after induction with DD1 at 40+13. My cervix wasn't ripe enough for a sweep until 41 weeks, after which I had 3 done , and eventually my waters went at 40+16 and DD2 was born that night.
I spoke at length about my wishes with the consultant midwife from 36 weeks. She explained that the risk of stillbirth doubles after 40 or 41 weeks (sorry, can't remember which) from 3 in 1000 to 6 in 1000 iirc. Because the outcome is so horrible, they choose to intervene with induction to avoid stillbirth due to placental failure. I took this on board and decided that 6 in 1000 is still a very small risk. I had to go for extra monitoring at 41+6, and was due to go again at 42+3. I still had to keep putting them off about setting a cs date, which they were keen for me to do.
My main aim was to avoid a cs, which I felt induction would increase the likelihood of. In fact I ended up with a high intervention VBAC, but for me it was much, much better than my induction experience. I laboured naturally (although weirdly, not text book at all!), my waters broke themselves, but when we went in for monitoring we found DD to be in distress so I was prepped for a cs. However it turns out I was fully dilated so she was turned with ventouse and pulled out with forceps. I had a spinal so didn't feel a thing . There was a point where I wondered if I had been foolish hanging on so long when things were looking dodgy, but the placenta was in good nick and her distress was probably caused by the cord round her neck.
Life's a lottery of risk calculation, but I prefer to be informed and make my own decisions rather than told what to do . Good luck with your decision.