You haven't supplied enough detail in this case for peple to advise - the depth or severity of her condition, why she's made this depression, etc, also assuming she is the day to day care giver for the children why are you questioning her capacity to care for them now? I would be very careful about playing the mental health card as it could prove detrimental to his continuing relationship with the children's mother. The move may impinge on the father's access to his children but it doesn't necessarily mean that it's a bad decision for either her or the children.
Seek advice from a solicitor who specialises in family law & custody issues and even though it goes against my better judgement to say this, you might want to think arguging that it would be detrimental to the children to be moved - moving away from their support network of family (extended?) & friends, their education may be affected, etc.