It turns out to be possible to persuade emacs and epiphany to play nicely together.
Sometimes Emacs presents you with clickable hyperlinks (in info documentation, perhaps).
To customize the browser in which these are opened, I am using the following in .emacs:
(setq browse-url-browser-function 'browse-url-generic browse-url-generic-program "gnome-open")
There are also specific epiphany-related functions, but I'm using the default gnome program for the moment. This can be configured via the default applications dialog.
When you download a text file, or use 'View > Page Source' in the menus, epiphany uses the gnome desktop's handler for the 'text/plain' mime type.
The simplest way of configuring this is to run nautilus, right-click on a text file and choose 'Properties'. Then the 'Open With' tab lets you select an application. Emacs should already be listed, or you can add it if not.