Child support is due whatever the circumstances. The CSA will take into account any mortgage payment the non resident parent makes as long as they have no financial interest in the property.
When both parties' names are on the mortgage both parties are liable for repayments and the lender will chase both parties for any arrears in mortgage payments.
However, it could be said if one party enjoys the sole use of a property they should pay the other party's share of the mortgage in lieu of rent for occupying their share of the property and the other party has their own living costs to find.
There is a duty for spouses to maintain each other and if the spouse living in the property is then left struggling financially and the other spouse can afford to contribute the courts can award them maintenance. It usually makes sense to sit down together (possibly with the help of a mediator) do some number crunching and come to some arrangement.
It would be different if the parties weren't married.