... I wondered if a record was kept of those baptisms carried out by hospital staff. ...
Hmm. An interesting question. But I'd think that I'd precede it with asking whether,
in reality, such baptisms were ever done. I know that
in theory such a thing could happen but I have my considerable doubts whether anyone in (say) a 20th century hospital would ever do such a thing - "Not my job", "What if I got the words wrong?" are some of the responses that spring to mind. (I'm tending to assume 20th century here because of the reference to hospitals...) And if it did happen, as it
might with more old-fashioned staff if the chaplain couldn't get there in time, why would the hospital record it? That implies a commitment on behalf of the hospital to religious matters.
I wouldn't be at all surprised if it happened in (say) the 19th century with midwives attending home births - the effect of the religious revivals of the time and no available chaplain.
Don't get me wrong - I don't want to deny it
ever happening - it's just that if there are no records visible, my gut reaction (which can be wrong) is that it's because there were no records in the first place. It's certainly worth an enquiry...