I work in the NHS as well, but with adults, in a fairly Welsh area. Not a single Welsh speaking patient has shown any hostility to my learning or speaking Welsh.
Among non Welsh speakers there are two responses, one of which I've only once met in the hospital situation (from a colleague). She tried to tell me that Welsh wasn't a proper language and that it would be a waste of time to try and learn it. This was before I started to learn the language in any formal way. The speaker was from Australia.
The other, much more common response is, I would love to learn it, or I started to learn it and it was very difficult so I stopped.
I remember someone telling me a few years ago, who had learned Hebrew in Israel, All Language Learning is Political. This is very true.
I have a T shirt which exemplifies this. I wore it at the Eisteddfod. It bears the logo
"I speak Welsh. What's your Superpower?"
But I do recognise that I have to be careful where I wear it. In a totally Welsh environment it's OK. (The Eisteddfod, for example). In a Learner Rich environment it's totally OK. In a totally non Welsh area, London, or further abroad, there is no problem.
Where I do have to be careful is on the streets of Carmarthen. People have amazingly strong feelings about the language. Do they feel guilty because they don't speak the language? Do they feel they are being attacked for not speaking it?
And when I get my next uniforms they are going to have Welsh embroidered into them. Scary or what?