I feel terrible wondering about all this... but it's hard not to isn't it?
I personally think that whatever is wrong with Kate, it's serious. Either she genuinely was having relatively routine surgery but it went pear shaped (hence why they only announced it later) or there was something quite serious in the first place or some combination of the two. I know on Twitter lots of people think she' being treated for an eating disorder but a) I don't think the palace is stupid enough to outright lie about stuff and b) notwithstanding the many people who think this, both her and her sister (who often also gets accused of an ED) have always appeared to be extraordinarily fit and healthy and strong to me.
But if it is serious, William could need to be there for all kinds of reasons from the children struggling with it (and no matter what, I think it's pretty clear that both W&C make a big effort to be hands on, emotionally invested parents so nannies or no nannies, their priority is the DC), to her having a bad day etc.
Of course, personal could also be something happening to one of the children or even outside of his family - his in laws, a close friend or similar.