I had an induction at 37 weeks due to pre-eclampsia. I was already an inpatient, so had a cannula put in the night before. I don't know when one is put in if you're coming in from home.
Day 1. I had 2 sets of the gel pessaries (6 hours apart) to try to kick start labour. One hour of baby CTG was needed prior to each gel. Other than having to return to the ward for hourly monitoring, I was free to do what I liked. You're encouraged to move, so I went for a walk around a nearby field. Triggered a few mild contractions, but nothing major, and certainly not labour. I was re-admitted to the antenatal ward at 9pm to get some sleep.
Day 2. I was bought down to the delivery suite at 1am. 3am I had my waters broken. Since that also didn't trigger labour, at 5am, I had the drip attached, which definitely did. Managed the pain on gas and air until 3cm, so had some morphine. Five wonderful hours later, where I'd slept soundly between contractions, I was 10cm and ready to push. 20 minutes later I was holding my beautiful baby girl. No interventions, and a small second degree tear.
I can't really comment on how likely your induction is to be delayed. It's dependant on how busy the delivery suite is, how many others need inductions that day and how medically necessary it is.
I hope all goes well for you. Try not to worry. Read up on induction, so you know what to expect.