I haven't completely solved this problem, but I have found more details and a workaround. The gist of it is that doom-themes  appear to use the same face for the header-line as they do for the modeline, causing this visual confusion. There are several tickets open on this:
The workaround as per ticket #320 is:
(face-spec-set 'header-line '((t :background "#23214b")))
However, as you can see from the screenshots above, prelude - or my prelude config, not sure just yet - was doing something clever and setting up this face to a more sensible value. I will continue investigating, but this at least makes the theme usable.
Found a much better solution to this problem. On the Emacs with prelude, I did
customize-face for face
header-line. There you can see all the properties for the face. I then did
customize-face for the same face using my new configuration file, and replicated the configuration I had in prelude, and saved it to my custom file. The generated lisp is as follows:
;; custom-set-faces was added by Custom.
;; If you edit it by hand, you could mess it up, so be careful.
;; Your init file should contain only one such instance.
;; If there is more than one, they won't work right.
'(header-line ((t (:inherit mode-line :background "#2b2b2b" :foreground "#f0dfaf" :box (:line-width -1 :color "#585858" :style released-button))))))
I could't quite replicate the grey colour on the border of the box, so
#585858 is an approximation. However, the final result is very close:
FWIW, the options in custom were as follows: