KP issued the photo to media houses saying it was taken last week by Willy.
It's a lie. It wasn't taken last week nor was it a "photo snapped/taken". It's an amalgamation of at least two photos and not a depiction of last week.
As much as the British Media are happy to kowtow to the RF this isn't the same for international media whom they sent the photos to. If it were the Royal Rota, they would've gotten away with it.
The AP is a reputable news agency, not a tabloid or general broadcaster. They provided news for general media.
KP can't produce the original photo cos there isn't any such and they can't hurriedly put one together cos Kate's face can't be shown for at least another week. So AP had to issue a "kill order" as it was deceptive narrative they'd been given. In fact the AP could sue the RF for attempting to destroy their business if not for the fact that Kate apologized publicly.
Mother's day photos will usually have her engagement ring showing but even that is missing. Kate lives in Windsor with the children so that doesn't preclude her from taking a photo with them.
Kate's undergrad thesis was in photography of children and the Victorian era.
So for someone who studied photography in university, who has been praised as a fantastic photographer by most British papers to amalgamate different photos into one as a "proof of life" news update from her royal office staffed to the hilt means it was simply impossible to get a recent photo of them together.
Ofcourse there're loads of photos of Kate and the kids but foremost on their mind is that she can't be seen yet but they also wanted to continue deceiving the public she's well by way of a recently taken a photo so they created one digitally.
The king decades older, dealing with cancer on the other hand is well enough to record a video message for the commonwealth service.