The age of the child matters a lot here. Children under 16 need consent from both parents to get a passport. It's technically possible for one to sign off and not appear but I've heard that can be a pain. https://travel.state.gov/content/tra.../under-16.html

A child 16 or 17 and who has their own ID can apply for their own passport with proof of parental awareness.

It may be faster to get a passport in the UK as passports in the US are getting processed slowly - 3-5 months, even with expedited processing.