"Have successfully applied" indicates that OP has applied for and received the kids US passports.
US citizens are required to enter and leave the US using their US passports.
If the kids are eligible for UK citizenship they need UK passports to enter and exit the UK.
Mother has file US income tax on an annual basis ans needs the kids social security number for child credit.
At point of entry I personally would use the citizens line for the whole family. Nobody expects a parent to manage two small kids when a second parent is available.