OP, does your church have any kind of organized activities for children (Sunday school, "children's church") during Sunday worship? I'm asking because (a) I'm not familiar with the Anglican church, and (b) I was raised Catholic, and we didn't. Children went to Mass with their parents, and religious instruction was held after school or on Saturday mornings. It sounds as if what Church Mom wants is the equivalent of the sort of kiddie activity many Protestant churches offer during worship time. And there's certainly a place for that, and she's free to organize it, but she shouldn't be hijacking an activity that uses church space but is understood to be secular. There are 24 hours in a day and seven days in a week -- she could easily find time for such a group that doesn't impinge on your existing play group.
I belong to a dance group that practices Wednesday evenings at a church. Other activities go on that night in other rooms. Some are church-related (Bible study, committee meetings), and others are not (chess club, Scouts). If the church needs our practice space for an activity on a Wednesday night, they get priority, but that only happens twice a year, and we're happy to take the week off or find another venue.
But that isn't what's happening here. It's one person trying to hijack an activity that has nothing to do with church and turn it into a church activity. I'm glad she backed down, and I hope she finds a time and place to hold her activity that doesn't impinge on yours.