I haven't been to hospital for years, so I have no experiences there, but I've noticed that older doctor's assistants, nurses and pharmacists tend to have problems figuring out my situations due to discrepancy between legal gender and my name/appearance. I found the following conversations especially entertaining, though also a bit frustrating:
At the GP's office, going to turn in a urine sample my doctor requisitioned for me around a week before:
Me: *Hands over the requisition* "Hey, I'm here to take a urine test, here's the requisition."
Doctor's assistant: "Oh... hm... we close in just thirty minutes."
Me: "Yes, but peeing doesn't take thirty minutes."
DA: "Hm... let me see..."
DA: *Spends two minutes looking from me to the requisition to the computer monitor repeatedly*
DA: "Hm... but girls are supposed to take this test intravaginally, not as a urine sample!"
Me: "Oh, but you see, I have a penis, so sadly I have to take a urine sample nevertheless."
DA: "Oh." *Thinks very hard for a minute*
DA: "You see, I wondered a bit, the Person ID number on the requisition is male. Does this mean the doctor didn't fill in the wrong ID number?"
Me: "Yep, sadly I won't get that changed until next year probably."
DA: "Oh! Well, you can just take the urine sample then!"
In pharmacies where I live, which is a bigger city with many trans people, most pharmacists are good at figuring out discrepancies. But recently I went to a small-town pharmacy while visiting my sister, and had the following conversation:
Me: "Hey, here's my ID, I'd like to have my prescriptions for [medications] filled."
Pharmacist: "Okay, checking your prescriptions now." (They're issued electronically)
P: *Looking from me to my ID etc again, for minutes.*
P: "Hm... This must be wrong. The names on your ID and in our system are different!
Me: "Ah. You see, I changed my name earlier this year, and apparently in some pharmacies it doesn't auto-update from the census registry."
P: "Yes, but... this must be wrong..."
Me: "Well, if it begins with a [letter] and ends with a [letter], it was my old name. I also changed my last name, the old one was [old last name]."
P: "But... it says [old first name]!"
Me: *cringe* "Yes, that was it."
P: "Oh. I thought your driver's license had got the person ID number mixed up."
Then she spent fifteen minutes trying to change my name in the system, didn't figure it out, asked if it would be okay to have my old name on the medicine labels just this time, and I told her that I'd rather go to another pharmacy if that was the only alternative. In the end, a younger pharmacist chimed in and told her that the name would update when she printed the labels, which turned out to be correct. So at least now I know that if I ever go to another pharmacy in the same chain, I can tell them what to do if they don't already know.
At least these situations are kind of innocent and entertaining (although frustrating since I get tired of spoon-feeding my history to clueless people), and I can only remember being misgendered in a health facility once, by a pharmacist. Other than that, most of the time people figure out the situation themselves, don't give it much attention and are very appropriate and respectful. I think the other posters are right though, that there is a difference between age groups and how much they have been taught about LGBT people in their education. But many of the older health workers also seem to be knowledgeable and respectful, but then they may have had additional courses (I'd think some institutions/employers focus on LGBT awareness and make sure their employees have sufficient knowledge), or personal knowledge through friends, family or simply interest.
