Some churches operate Mainly Music programs for infants and their mothers (typically). Its a good community service in itself, but a church has a mission. The MM program has to be part of that mission.
I've encountered churches where the mission connection is known, but not handled well. MM numbers drop after people are button-holed to come to an Alpha course, or even bring their child to Kid's Club on Sundays.
That's too fast.
There needs to be a step-up of involvement, not a plunge.
A couple of times a year invite attendees (mothers and/or fathers) to a function, maybe an early dinner or coffee/desert (timed for a young families' needs) where a Christian expert in some child-related discipline (medical, psychology, education, etc.) gives a light but informative talk about their experience.
Once a year have a dinner (with creche if needed) for parents to take a break, meet other parents, have a similar talk to above, but the person reflecting on their Christian motivations ('I' not 'you' reflection).
Offer mothers to come to a monthly 'support' group with a coffee and cake, a 'low key' expert available for discussion.
Those mothers who really like it might be invited to a perhaps monthly, or bi-monthly group were 'we talk about the deeper things of life with children, their future and how we use the wisdom of the Bible to help them and us....light discussions in Proverbs might be nice start.
All this is about making a pathway to Christian faith, not forcing a plunge into an icy pool.