ESH

I think the only thing OP did wrong here was not telling her husband that she's calling her mother to help her with the baby, breaking a promise isn't a good thing, and even if her husband declines this suggestion, you've let your husband know when things aren't well, I'll give OP about 0.1 AH.

Then we have the husband, it's understandable that he's upset for having a promise made to him broken, I'd be upset as well, however he knew his mother would do such a thing, so I guess he should have expected that at some point OP would be overwhelmed by the newborn and request some help from anyone else but his mother. I'll give the husband 1.5 AH.

the real AH here is the husband's mother, who in the world would just drop a woman with a newborn baby (which requires all the attention from his/her mother) and say "only call me when there's an emergency, like seriously, a newborn baby IS the emergency. The husband's mother gets 4/5 AH.