Yes, induction (anything more than a cervical sweep) would be on a labour ward.
They start you off in one of three ways - if your cervix is less than 2-3cm dilated, you will be given prostaglandin pessaries at around 6 hour intervals to soften the cervix and start contractions. After this, or if you are 2-3cm dilated on admission, they will break your waters (ARM) using what looks like a big crochet hook, hoping this will get contractions going in a couple of hours.
If this doesn't work, you'll be hooked up to a Syntocinon (synthetic oxytocin) drip which will make contractions start pretty hard and fast. There's no release of endorphins (your natural painkillers) with this and no gradual build up of contractions so a lot of women get an epidural. With the drip, you'll be advised to be continuously monitored which makes moving around pretty bloody difficult (regardless of the epidural) and being stuck on a bed can make labouring more difficult as you're working against gravity.
Here induction is suggested at 10 days, I went in at 14 days and was actually found a space on the labour ward for ARM at 15 days over and DD was born at 16 days over.
The risk of stillbirth does increase after 42 weeks but only from one tiny % to another tiny % and induction does carry it's own risks to mother and baby. It's entirely your decision even if it's not presented to you that way (eg you may be told 'our policy is to induce at 12 days over' but they won't tell you that you aren't obliged to follow their policies!)
If you don't consent to induction, you should be offered expectant management which are regular (daily/every two days) scans to check placental function and amniotic fluid levels and if there are issues with either of these then you can talk through your options with a consultant. As consent for induction seems to be the norm, some women do come up against problems with midwives and consultants not being wholly supportive of their decision not to induce (others have very supportive medical staff)
If you are worried, stressed or anxious though then this can delay the onset of natural labour, so try to relax! No-one stays pregnant forever, your baby will come one way or another :)
The main thing to remember is it is your decision - not the midwives, not the consultants.