Allow yourself to grieve. You are a person with feelings and the loss of a child at any time is difficult. When I worked L&D, I dreaded the demises, but it is a part of the job, and may be the most important part of the job in supporting those parents.
When I was 32 wks pregnant with my third child, a staff member from my OB's office was 36 wks. She came in one morning when I was working because the baby had been really active the day before, but hadn't moved for about 12 hrs. We thought (wishful thinking perhaps) that we found the FHR, but MHR was elevated and we couldn't be sure. The hardest thing I had to do was go in with the OB after the US and tell her her child was dead. I cried before I went in- another nurse offered to relieve me so I didn't have the stress. I left it up to the pt, having me (pregnant belly) or she could have another nurse. She chose me. She didn't deliver during my shift. I cried all that evening and woke up the next morning crying somemore. I wanted to call off sick. My husband, knowing how upset I was, supported me, but said, "is she expecting you to come in?" Answer was yes, and he said you need to be there for her. So in I went, crying the whole way, praying the whole way. She had delivered during the night, and they had done all the bath and pictures so I wouldn't have to. I took care of this pt and her family, crying with at times and supporting at others. They thanked me as they left, and wished me luck with my pregnancy.
Moral of the story- someone needs to be there for these pts- if you're upset it means you're human and you're feeling their pain also. It's not easy to cope with, but talk with your coworkers, manager, grief counselor- you will find ways to cope. You are being given an opportunity to nurse someone at the time they need it most- seize it. Just remember to take care of yourself also.