I do find it bizarre that The Times wrote a whole story on this. I mean, as a royal watcher it's vaguely interesting but I have to admit that I hadn't really thought about it much and if I'd been asked, I'd have just assumed the children were Sussex already so it seems a non-story to me.
The Mountbatten-Windsor thing was, I think, linked to Prince Philip. The idea being that his non-HRH grandchildren or descendants would take his name as his actual children couldn't. But there aren't actually a lot of them around. So Lady Louise (Edward's daughter) is Mountbatten Windsor. I guess technically Beatrice and Eugenie could have been too but they took on the HRH Princess E/B of York which is now what Harry has done for Archie and Lillibet. If Eugeuni or Beatrice had been a boy, even if they'd been Prince Benjamin of York, that imaginary boy's children would have been Mountbatten-Windsor.
I think Edward's grandchildren on James' side will be Mountbatten-Windsor (except, possibly, the first born who would have a title as he'd be an hereditary Earl?)
Ann's children also didn't get titles but they took their father's name hence they're not Mountbatten-Windsor.