A “notarized agreement” doesn’t mean anything if blindly following it is going to put your child in danger.

No family court (I hope) would agree that you should try to live in a car with your child all winter and be bouncing in and out of shelters rather than stay with your family. Especially since the father is backing out of a former agreement to let you live with him. Although if your son’s father wants full time custody and has a stable home situation, they may decide it’s better for the kid to live with his dad.

If you and your ex can’t come to an agreement, you probably will need to talk to lawyers in both MA and the UK, and maybe petition a court here in MA to set up a formal custody agreement.