"Wild Mountain Thyme" still gets done a lot N of the Border, and "Wild Mountainside" (an entirely different song) is catching on -it was sung at the opening of the Scottish Parliament by Eddi Reader (as, of course, was "A Man's a Man for A' that" by Sheena Wellington). Other Burns songs ever present.
"Follow the Heron" seems much less known S of the Border, but is very popular up here, both in clubs and with community choirs.
Tine Wears Awa'
Braw Sailing on the Sea.
Fields of Athenry can be a difficult one, as it has been inextricably linked to Celtic football club, and you are well advised to check the local allegiances before attempting to sing it in a pub session (sad but true).

Most of the ones mentioned above, I have only heard when "down south", and have to confess, that I DO like "When All men Sing" (maybe haven't yet heard it often enough to get sick of it!