I can't believe how rude these responses are. This is a new mum, who has given birth during the pandemic, with a newborn baby that isn't gaining weight as it should. She's under a lot of pressure, she has a lot to worry about, and you all sound ridiculous and nasty piling on her like this.
It would clearly be standard practice for a nurse to wash their hands before doing this. OP is not out of line for expecting that. It's basic hygiene and we're living in a pandemic.
OP, you're not unreasonable. Your baby is unlikely to have caught anything, but the nurse should know better.
If you feel up to it, you could contact the centre and say you noticed she forgot to wash her hands and that it's made you anxious. They should know that, their job is dealing with new mums (including stressed ones) and they should be mindful that things like this are important not just to prevent the spreading of germs but to ensure people who come there can have confidence that the standards are high in general and that there is attention to detail.
Otherwise, take care of yourself. Your baby is unlikely to have any adverse consequences from that.
Weight gain stuff in a newborn is scary but remember that babies are tough and they all grow at different rates. Look after yourself