Hi, Mick - I have a tough time with weddings, because I often end up singing music I don't particularly like, or that is not particularly appropriate for religious services. I don't ever want to get stuck singing "Wind Beneath My Wings." If I really hate a song, I just say I don't know it.
A Celtic wedding might be an interesting challenge, though. I don't think there's much music you would call "Celtic" that's in use in the Catholic church in the U.S. The only ones I hear often are "I Heard the Voice of Jesus Say," (tune: Star of County Down) "Be Thou My Vision," and "Danny Boy" and various lyrics to the tune of "Danny Boy." None of these apply very well to weddings - but you could use the tunes very nicely. The "Celtic Alleluia" is nice, but it's usually used in a little snippet before the Gospel, and it's over very quickly. Here is a link to sound clips from the "Celtic Mass," which Oregon Catholic Press published after the success of "Celtic Alleluia." It's kind of nice, but hardly what you'd call traditional Celtic music - neither is the "Celtic Alleluia," for that matter.
If you want me to, I can ask a couple of Irish priests what they do back home, but the priests I know aren't really "up" on music. I think you'll probably get better advice here, but there hasn't been time for a good response because Mudcat was on vacation this weekend (or something). It will be interesting to see what comes up in this thread.
I might suggest an Ave Maria, because it always seems they want one - how about the Gregorian Chant version, a cappella or with a harp or guitar? It doesn't have the heavy melody that the "usual" ones have.
As a seminary-trained church musician, I'd like to put in a request that you try to pick music that's appropriate for a religious service. "Mairi's Wedding" is a great song, but it would be better to do it before or after the Mass. It would be a great recessional march, as you suggested - after the priest says "The Mass is ended, go in peace," you can sing whatever the hell you want.
Hope that helps, Mick.
-Joe Offer-