http://web.archive.org/web/20140402131524/http://www.bbc.co.uk/news/uk/
Below the "News UK" there is nav bar. There is a border on that nav bar between items. Not only i cannot find a way to replicate it, i cannot find any trace of it in the developer tools.
I even looked for the hex code of the color in all of the CSS files for no avail. Why is that? Where does that border come form?
If you are meaning the archive.org link you reference, rather than the current live version of the site, then it looks like you may be referring to the #nav a which has a transparent background image. That image is a 1x10 vertical line.
It shows on each a tag in the nav bar list, and refers to skin.css line 47.
If you mean the actual live current version, it is on .navigation-wide-list__link > span as a border-right and shows in core.css on line 5169.