Hi, and yes!
I was diagnosed with OC in my first pregnancy, with similar timing to you. It will depend somewhat on your bile levels, but actually it turns out that different hospital trusts have different policies on OC treatment, so you may have more options that you think. I was at a hospital where they don't induce as a matter of course until over 40 weeks. Apparently being induced at 37 weeks by default is NOT necessarily evidence-based. You may want to question your consultant about this. I was however told that it was the norm at many other hospital trusts (indeed, now pregnant with #2 and in a different city, I've been told I 'won't be allowed' to go over 37 weeks!).
Definitely do take up the regular bloods, monitoring and growth scans as these are reassuring and helpful indicators. Have a look at the bile acid levels. If yours are high, you may feel positive about an early induction. If your bile levels are relatively low, though, you may prefer to refuse an induction until your due date.
There is no actual evidence showing that the urso medication has any effect, but there's also no evidence that it does any harm, so I decided to take it just in case!
I was obviously anxious about the increased risk of still birth so I did question the lack of induction, but felt confident that the consultant was knowledgeable and that their policy was evidence-based. This has made me sceptical of the 'must induce at 37 weeks' brigade! It obviously isn't as cut and dried as some consultants make out. You may want to contact some trusts that have a different policy to ask about why. I did however ask for sweeps once I was over 38 weeks, and gave birth at 38+5 after labour started naturally.
I started out having a home birth, but transferred into hospital when my waters broke (at 9cm dilated!) and there was meconium in them. I also had an anterior lip on the cervix which was slowing progress, so midwife suggested hospital and I trusted her. Ventouse birth at hospital, having done a bit more pushing to get the head into the birth canal. Healthy happy baby. Meconium in the waters is quite common with OC, apparently. Many OC babies also come a bit early naturally.
I will have bloods taken regularly in this pregnancy to see if the OC develops again (it recurs in 40-90% of women who've had OC in a previous pregnancy). I have also tried to cut down on fats and sugar, drink extra water, and I am taking a herbal supplement called milk thistle that is meant to be good for the liver.
If all goes well, I will be refusing an induction at 37 weeks and going for another home birth unless there are other risk indicators (like very very high bile levels). I feel that induction is itself a type of risk, and I don't believe that there is clear evidence for it beyond 'lots of trusts do it, so we will'. I would however probably have an induction if I went over my due date or if my bile levels rose dramatically. And I will have sweeps again from 38 weeks onwards.
Hope this is helpful!