In the iOS Mail app, how do I share an email, or save to Files, etc.?
As far as I can tell, this is the best solution:

Tap the button to Print the email
Pinch-zoom the print page preview to fill the screen
Now you should see what's effectively a PDF of the email, including headers and all pages, and there will be a Share button in the top bar
Tap the Share button to get all the standard options to share a PDF of the email

If the email has any attachments that can't be displayed inline, then it looks like the PDF that's shared will just include an icon representing the file. So these attachments would have to be opened and shared separately.