I'm a nurse that works in a long term care facility. So not a hospital, but I can't imagine there would be too many differences in experiences. I'm not full time yet, and not technically out at work, but I do wear nail polish, earrings, women's clothing, etc, and have on occasion went to work in full makeup. Like, I had a meeting on a day off this week and I don't actually own any men's clothing any more, and I got more comments on my sparkly nail polish (they're all jealous of my nails xD) than the fact that I was wearing women's clothing.
About 6 months into HRT rumors started appearing that I was transitioning. People don't say it to my face, but there is gossip about me. Pretty much everything I've done was met with compliments to my face, but gossip behind my back. When I started growing my hair out, my ears pierced, my eyebrows done, started wearing women's scrubs, then women's shoes... everything. I've heard a boss say that they need to step up their game because my nails looked nicer than theirs, or that I had the best eyebrows in the building. I haven't had anyone be ugly to me about it though. In the beginning the gossip bothered me a lot, but I got over it. I get ma'am quite a bit accidentally from coworkers, and from those who don't usually work with me or know me before transition.
From patients it's been weird. Before transition it was pretty common for patients to call me doctor, which was annoying because I'd have to correct them. Most of those same patients now call me baby, sweetie, honey, or some other petname. I've had patients ask me if I'm a man or a woman, most of my patients call me ma'am now or change pronouns spontaneously, I've had a couple male patients ask me to hold their penis for them while they urinate (like wtf seriously?), I've had a male patient halfheartedly try to pull me in bed with him. I mean tons of stuff all related to being perceived as female rather than male, not necessarily trans stuff.
Overall it's been a pretty easy transition for me, but I'm not planning on officially coming out until legal name and gender change, which I'm planning for the first quarter of next year. So no idea how it's going to go once I socially transition at work. I am out to about half of the coworkers I work with regularly, and everyone is super okay with it.