How to prevent losing music and playlists when switching from individual to family Apple Music account?

According to the Support Article from Apple that pertains to that exact circumstance of adding a member to your Family group when they already have an Individual Apple Music subscription.

• If someone in the family group already has an individual or student subscription to Apple Music, renewal of that subscription is automatically canceled when the family organizer subscribes to a family subscription.

Get an Apple Music family subscription - Apple Support

There should be nothing else you need to do and the email your wife received sounds like a standard form email due to here individual subscription being cancelled. Even given that, I would still be inclined to contact Apple directly for confirmation and you can do that here:

Contact Apple Support - Apple Support

It should switch her over to the Family Subscription at the end of her current subscription period. You do not want to cancel the subscription manually as that can cause library and playlist loss.

With that said, why did she not pay for hers? If it hasn't switched, the subscription should still be charged normally to her payment method. Once in family sharing that charge would have been charged to your payment method.

The fact its warning about lack of payment is concerning in many levels as it generally means it's already tried to be charged but failed for some reason and it has not switched to the Family subscription you shared.

What exactly did you do when you "put her inside your family"?

Can she see the Family option in Settings on her device?

Can she then see the shared Apple music subscription in there?

Can you see that the subscription has been shared with her in Settings on your device?

click here ➜ Add a family member to your shared subscriptions - Apple Support