My DD cut her leg badly on the beach, i just grabbed the first thing i could out of her bag and that happened to be a pair of spare knickers. In these sort of incidents it is not about being sterile, obviously you are not going to grap the toilet cloth but babywipes should have been fine. The most important thing would have been stemming any blood flow and calming DS down. The time for sterile is when the wound is being cleaned up ready to be closed, either by stitches or other method. The mouth is a hotbed of bacteria, so an unsterile babywipe is really not an issue.
I think the OPs concern is valid though, i would be concerned if there was no one first aid trained on site? Am i to understand this to be the case? The thing is, children split their lips all the time, so maybe the nursery underestimated how serious it was? They sound like they did the right thing, called mum in and i imagine you took him off to A+E? I'm not sure what they could have done differently really or would have done even if they had a doctor on site tbh.
When my DD cut her leg, i was in two minds whether to take her to minor injuries, i think the only reason i did is because i had nothing with me to clean it properly and i had to walk right past it on my way home, otherwise i would have dealt with it at home, as it was, i think i made the right call as it had to be glued and steri-stripped, but to be honest, i really don't think it would have been in the end of the world if i stuck a plaster on it at home tbh.
I think the OP reaction was perfectly natural, you can't help to feel angry if your child injures themselves in someone elese care, if they are with you, you feel angry with yourself. But in retrospect, i think that i would ask to see the accident recorded in the injuries book, i can't imagine this would be otherwise as they called you in.