Not on a third baby, but both my two pregnancies were 37 week induction for pre-eclampsia. Pretty sure I've already mentioned my experiences in one of your other threads already, but I'll run through them here too.
Both babies I started with a low Bishop score. Wasn't expecting it to be anything but though. I had gels to get things started with both. First because it was standard at my trust at the time, second because baby was positioned too high for a balloon. A single gel got me to the requisite 2cm for ARM with my second, so hopefully it will be the same for you.
No signs of labour with either, so then it was a matter of waiting for a bed on delivery suite. Be prepared to wait, as triage comes into play here. I had to wait 4 days for a bed with my second due to how busy it was. If things had happened like: contractions started, waters gone, or baby and/or I had started showing signs of deterioration, then naturally the process would have been expedited, but none of it did, so I was kept in the holding pattern.
Had ARMs with both. Nothing happened, so I had the drip. As it was gradually cranked up, so did labour. Maybe not as slow as natural labour, but that hour or two before strong contractions set in helped. Since I've no regular labour to compare to, I wouldn't know if the pre-eclampsia affected it. You'll have continuous monitoring, but hopefully your trust will have wireless too. I spent the first two thirds of labour with my second trundling around my room (drip in tow), and trying out different positions next to the bed. Very different to my first where I hated everywhere but my bed. You'll have frequent BP checks too.
As for afterwards? Well, tired obviously. Delighted, happy. Again, no natural labour to compare to, but I doubt I would have felt any different. And it was the same feeling for both babies, even though they were born different times of day, and via different methods (DC1 vaginal delivery mid afternoon, DC2 em. C-section (baby got stuck, nothing to do with pre-eclampsia) middle of the night). I felt I went into labour faster, and coped better with my second. I was just one of the unlucky few whose second delivery was not quicker or easier.
I then stayed 2 nights postnatally with both for the blood glucose checks for baby, and to keep an eye on me. My BP immediately went back to normal, and I felt like I otherwise had before, albeit a bit sore in my vagina/abdomen. I had meds for a couple of weeks after my first (weaned off quickly), and didn't need them at all with my second.
Always happy to answer any questions.
Wishing you the best of luck for your induction.