I think most people wouldn't advise someone so young, at the most vulnerable time in their life, to move in with their boyfriend.
If your relationship is good with your Mum, you'd be best placed to stay with her.
Having your first baby is a shock for anyone, at any age. Even easy babies are a shock. It's the 24/7 of it and most people under 20 need a level of support. Or rather, outcomes etc are better if they have support.
Your BY's Mum might be thinking, if you're old enough to make a baby, you're old enough to have your own place. But that isn't necessarily be the best thing for you.
You don't know someone until you live with them and both of you are still growing up and may grow apart. You may not, my DD has been with her DP since they were 17, they are in their 30's.
They didn't have their own place until their mid 20's though.
This forum is very anti families, but extended families can work well.
The pair of you need to tell his Mother that the decision has been made.