Not really an emergency, but something I had actually been worried about - baby passing meconium.
It was something I always knew I was at a higher risk for, since I have long gestations, so I'd done quite a bit of reading about it, and ultimately came to the conclusion that intervention was rarely helpful and that I would assess the situation rather than immediately transferring for it (which I couldn't have done anyway as it turned out).
I was having some mild contractions then had a little leak of slightly brownish coloured water (that was my water breaking - don't think I had too much left at that point - this was 43+6 weeks) so I knew that I was likely dealing with mec. Checked DD's heart tones with my little Doppler thingy - she was doing just fine, and moving well, so I figured I'd wait and see. Ended up having a super fast labour, and she was out in about 45 minutes from that - totally covered in thick mec, but fine.
My mom caught her and handed her to me - she was alert with wide-open eyes, not really crying, just gave a little one then looked around. I quickly checked her mouth, which was clear - I was prepared to use my own mouth to 'suction' if necessary, but no need.
I have to say that having read quite a bit about it, and having a pretty clear plan in my head as to what I would do if it happened really helped me to feel calm and clear about it all. I had similar 'plans' for other potential emergency situations, like shoulder dystocia and PPH.
We did end up transferring into the hospital for observation for a few hours after the birth, completely for my DH's comfort - but I managed to keep DD2 skin-to-skin the whole time, and didn't have too much trouble refusing various things they wanted to do there - mostly by not putting DD2 down at all!