there is an amount of your income that is set aside for basic living expenses before child support is calculated - that varies depending on whether you are single or partnered. your partners actual income doesn't come into any calculation. so if you had a minimal income previously (income support) it was pretty much excluded from calculations based on basic living expenses, so now if you have no income (assumption!) it won't change the child support income

your ftb will take into account yours and current partners income

you will still need to make contact with both departments on some things (address change, change of circs like working etc) but not all things (if you have a shared care arrangement with your ex, that will auto-update between the two departments if it changes