If you can Anab put your feelings in a mental box for a little while, so you can get the practical life details sorted out.
You’ve had good advice here about getting and securing crucial documents, also, opening a new bank account, taking half of the joint account to fund it, and changing your passwords.
You can’t force your husband to leave the house, but you could ask him to give you some space while you process everything. If he is hoping to ‘win you back’ he might do that.
Don’t hide what has happened. This is not your secret to keep. Share with friends and family, you will need their support. Get your solicitor on board ASAP and take their advice.
His relationship with your child can be sorted later, via the solicitor. Don’t be too soft at this stage, let him worry. Does he really care that much if he can shag the child’s nanny in the family home?
If the nanny came from an agency, I would absolutely tell them, although that’s not a priority. She really shouldn’t be allowed to work in a position of trust ever again.
I know you are in shock, but toughen up for a little while to get through the practical stuff, it sounds like you have considerable assets to deal with.
Take care, and I am so sorry your husband did this to you. Unforgivable.