Ob/Gyn told me at my first appointment that I was high risk and would need to be induced up to 2 weeks before my due date. So, we just told everyone our due date. I was induced at almost 39 weeks, labour started instantly and lasted a day and a half - I wouldn't have wanted my parents or my MIL to be worrying about me for any or all that time. So DH gave them the good news when our precious DD1 arrived. SIL was pregnant too, due 10 days after me and she announced that she would be induced on a date before my due date, I said oh, that's great, you can show me the way and give me all your tips, I knew she was delighted to think she was going to have the first baby and I didn't want to burst her bubble, until my own bubble burst its way out!
Was due to be induced for DD2 also but she appeared early, my Mum minded DD1 so she knew but it was a surprise for everyone else. My Mum could be trusted not to say a word to anyone, I know my MIL's side would have been updated each other every hour even if it was just to say there's nothing to say about SageRosemary.
You are the most important person on the day and your partner needs to be able to support you fully, phones should be switched off or on meeting mode for your labour. Best of luck for a safe delivery and a healthy arrival. Your MIL is of course delighted for you and the arrival of a grandchild, maybe just promise her that she will be the first to hear your news, I think she would be charmed, she will probably be a doting granny.
It was the same when we announced our engagement, we already had the Church, Hotel, Band, Photographer booked for a date 5 months away - no one else needed to be involved before that.