A very typical hat from Queen Elizabeth (the Queen Mum, that is -- but not yet at this point) for her elder daughter's wedding -- I really sort of wish big, fluffy, yellow feathers would make a comeback in royal hattery.
As for the woman on the right of the photo in the dark hat -- that's the Dowager Marchioness of Milford Haven, born Princess Victoria of Hesse and by Rhine, grandmother of Prince Philip. She was a granddaughter of Queen Victoria, and she's the one who lent Philip the "Mountbatten" name after he renounced his Greek titles; her husband, Prince Louis of Battenberg, had taken the name Mountbatten and the title Marquess of Milford Haven after the family renounced their German titles during World War I.
I can give old Vic a pass on this rather strange, fedora-esque hat on this occasion, because she's in her mid-eighties, and people in their mid-eighties have a license to wear whatever the heck they want! (Erm, except Lilibet -- but she's not exactly your typical pensioner, now, is she?)
Most of the photos from this wedding are in black and white, but here's a little color version of this one to check on the colors of hats/dresses:
You can see that the Queen Mum's wearing golden yellow -- nice -- and Victoria's in what seems to be dove gray. Both very appropriate, I think!