Nappy - absolutely not ok.
Key Worker - why doesn’t she know? Doesn’t each key worker need to complete paperwork or tapestry after each day. Even if she was in a new room I would find this unsettling.
Snot - ordinarily no, my children are forever snotty even if I’ve just wiped their noses but against the backdrop of the other issues it doesn’t paint a great picture.
Definitely have a chat with the manager and keep a watchful eye. Trust your Mum gut. If it doesn’t feel right then it usually isn’t.