An overview of the replies

Thank you to all who offered advice for storytimes to this nervous librarian. The gist came out to be:

1) Remember children are not a particularly critical audience--if you flub, they probably won't know or at least won't care.
2) Practice and plan! Reading a book over and over outside of storytime until you get really familiar with it will help. You can read to other people's children, pets, and even turkeys (I particularly loved that one). Overplanning, so long as it doesn't get in the way of spontanaity and the ability to handle interruptions or other issues that arise, can be very helpful. So can having a set welcoming format.
3) Have fun and don’t worry about looking silly.
4) Use a puppet to put the attention away from you and onto the story.
5) Attend public library storytimes or conferences to get ideas.

I now realise that being nervous in a performance situation is completely normal and apparently common. I think in my case there are factors that make it more likely (a good dose of social anxiety, being an only child, and not having kids of my own). But I also have friends' kids to practice on and I have at least taken classes in children's literature and storytelling. I should clarify that our children are not so much sick as living with chronic conditions such as spina bifida, amputations, cerebral palsy, etc. Rarely do we have a child whose condition is particularly bad; in fact they're liable to run you down in the hall with wheelchairs, etc. Only when they've just had surgery or are dealing with a lot of pain would I say they wouldn't be up for a story. My biggest challenges in terms of a hospital library waiting room are that there are many distractions (including video games and toys), a varying number of children (sometimes just one, sometimes more, but never huge groups), and the fact that our wait time has gotten better, so they may have to leave in the middle of things. I am trying to do them at a set time (11 am on our biggest clinic days), but it may take awhile to find the best time, and that's something I can work with the clinic folks about. It may be better to do it at times when the most young kids are here.

Many of you commented on my job sounding great, and it is, I love it. The only drawback is that I'm part-time; trying to do the storytimes, book cart, early lit programme, and summer reading project on 20 hours a week in addition to my normal duties is going to be a challenge. But hey, now I have an excuse to get out of the hospital and visit some local libraries and watch how they do things. :)