The risk of the placenta failing increases from about 10 days overdue (possibly earlier in older women or women with diabetes), which can lead to the death of the baby. Different hospitals will have different policies about when to bring women in for induction. Often just because of the numbers of women, they will leave it until 40+12 or 40+14, unless there are other concerns.
I was induced at 41 plus 10, then again at 11, 12 and 13 days over. It was done at 10 days over due to suspected large baby, but took several days and repeats to work. Had I known it would have a possibility of being unsuccessful I would have waited longer. If I have another child, induction is not for me. Simply because I would rather be at home waiting than in hospital!