They will usually send you for scans to check your placenta is still working ok after 41 weeks or so. If you don't want to be induced you can demand daily scans and decline induction - "forget" to go the induction appointment if necessary.

My DD was 11 days overdue and when she came out my placenta wasn't "old" looking at all, she didn't have any signs of being at all overdue (wrinkly, no vernix, long nails etc.) and the midwife said she looked as if she'd come exactly the day she was due.

My sister was born at 44+1 (in the 1960's) and though the placenta was "gritty" and old-looking she was absolutely fine.