The Queen's own website can be considered the ultimate indicator of such matters. As usual, the website handles this situation very subtly, but clearly. Compare the way it describes the Duchesses of Glouchester and Kent.
HRH The Duchess of GlouchesterThe Duchess of Gloucester, formerly Birgitte Eva van Deurs, was born on 20 June 1946, in Odense, Denmark.
HRH The Duchess of Kent
Katharine, Duchess of Kent was born on 22 February 1933, the youngest child and only daughter of the late Sir William Worsley.
The Duchess of Kent's style is one typically reserved for a widowed or divorced duchess, however, she officially retains the HRH status. So, the Queen has consented to allow her to adopt this style, but no official change in her status has occurred. I'd call that quietly separated.
Given the circumstances of her religion, their royal status, and of course their families, it would not be surprising if the Kents remained married and quietly led separate lives. It happens with many couples.