Op's mother's boyfriend/partner was the one who 'had a go' not her own dh.
Her h has a right to question, or at least discuss, but as other have said, if one of the parent's isn't content to leave their child with whoever, for whatever reason, then it's not going to be a break for either of them.
OP here has a right to be heard, listened to and respected.
The mil hasn't done this, her boyf has waded in, and the H isn't exactly dealing with this.
It's a case of a 2yo that the mother doesn't feel is ready. That's about as cut and dried as it can be.
The fact that she's been placed under some considerable pressure to do something she's genuinely uncomfortable about doing is the issue here.
Day in, day out we tell each other here to listen to our gut, pay attention to our instincts and do what feels right.
Someone has to make a decision, and the mother gets the final word here. It actually doesn't matter if she is being silly/unreasonable or whatever, it's what she feels comfortable with, knowing the child she's raised.
If the IL make this an issue it will have consequences. They'd be best off trusting her judgement and backing off. It'd be the far quicker way for them to get what they want than forcing/haranging or pressuring her.
This situation seems all about them though and not at all about the child.