Yeah "cottage" is hilarious. But it was named in the Victoria era by toffs who never even entered their own kitchens. They probably thought it was positively quaint!
Speaking of kitchens, the kitchen in 1A is easy to find online, it has the famous big cooker hood designed by that arsehole Tony Armstrong Jones.
I don't know if they kept it or not, but that kitchen is a big ass catering kitchen. Apparently they also have a kitchenette for making snacks and tea.
I heard the Nanny doesn't live in, but rather lives over near Nottingham Cottage in the staff quarters by the garages that house a lot of the drivers and other household staff. Those places would be bedsits essentially. Though nobody really knows where she lives of course.