I apologize for not having screen shots. I may go back in game and get them later but don't have the time this moment.
I love cavalier hats (have a few RL, from my reenactment days) so was eager to get this bundle. Logged in on my main, a Redguard DK in heavy armor (mostly Mercenary). Her hairstyle is black close-fitting cornrows that hang down her neck. Both in preview and in the actual game, donning the hats removes her hair, leaving her with some dark stubble. I wasn't even going to get the hats once I saw that, but decided to test on other alts.
My male Breton DK's shoulder-length red hair shows under the hat, as does my female Altmer sorcerer's longer red hair. I have not tested on my two female Imperials (Templar with an elaborate up do in copper, Nightblade with black short tousled hair). I don't expect to test on my female Khajiit or female Argonian.
Fortunately my main isn't ditching her turban any time soon, but I bet some Redguard players would like not to shave their characters to get the hats on.