We used some non-denominational religious music - specifically, “Say a Little Prayer” by Aretha Franklin.
Then, as soon as I broke the glass…
Good loving! good loving!
Good loving! good loving!
I was feeling so bad
I asked my family doctor just what I had
I said doctor…
Hello Again, I’ll also lend you the Rondeau. It was not written for a church; it’s from “Symphonies and Fanfares for the King’s Supper.”
I don’t remember everything we used…I know Canon was in there, as was Jesu Joy of Man’s Desiring. The recessional is one I do remember, though: “It Had To Be You”.
If you think Clark’s Trumpet Voluntary is too stuffy and formal, you could go with the rock and roll version, Trumpet Volunteer (song starts at the 4 minute mark, but the interview’s worth listening to as well; Peter Sellers interviewing himself).
It’s a beautiful day in the neighborhood
a beautiful day to be neighbors
would you be mine?
could you be mine?
I walked down the aisle to Jeremiah Clarke’s Trumpet Voluntaru, played by our organist. It’s not specifically religious, and it sounded lovely. Our recessional was Charles-Marie Widor’s Toccata (Symphony No. 5). Some other secular pieces we had were Voluntary in G by John Stanley, Prelude on Rhosymedre by Ralph Vaughan Williams, Bach’s Minuet in G, which is one of my favorites, Allegretto (Sonata No. 4) by Mendelssohn and Elgar’s Nimrod from the Enigma Variations. They were all very nice pieces and included a couple of my favorites.
My husband’s suggestion was either Darth Vader’s march or Devil With the Blue Dress On.
Maybe it was Nella Fantasia, which is from Morricone’s “The Mission” and arguably The Most Beautiful Piece Of Music Ever Written.
Here’s the program:
Sigh, I wish it was me, but I would look silly in a dress!
We used Mark Motherspaugh’s Canon from the Royal Tenenbaums Soundtrack for the processional and Linus and Lucy for the recessional. And yes, we did dance up the aisle.
The Canadians on the board will appreciate this - my sister and her husband had the Hockey Night In Canada theme at the end of their recessional. 
(You really can’t have an organist in Canada and not play our unofficial national anthem.)
We used The Final Countdown.
My processional was Handel’sThe Arrival of the Queen of Sheba. It was what my mother had as her processional too, so for me it was a nice link to the past.
It’s 02:12, so it might be too long for you, or too fussy and baroque for your taste, but it is a very apt piece of music.
Sorry for the hijack: Siege I didn’t realise the big day was over and done , congratulations!
Well, there is that lilting, poignant love theme in 3/4 meter from the prequels. 
Tubular Bells
The Ch-Ch-Ch-Ka-Ka-Ka sound from the Friday the 13th series
The Theme from A Nightmare on Elm Street
although you really cannot go wrong with the Imperial March 
Isn’t there a piece call Luke and Leia from the original triology?
I personally am fond of Yakkety Sax.
No, I’m not joking.
You know, I’m a real traditionalist, but at the wedding described in the OP, I might just say, go for the Star Wars music. Either the Emperor’s March, or the Rebel March. It can’t hurt, and you’ll be pleasing the man who you plan to spend your life with. So for this one thing, let him choose.
There’s not going to be much of anything that hasn’t been heard before, so have fun. I went to a non-church, but religious ceremony wedding, and the recessional was “I Feel Good” by James Brown. Or further upthread someone suggested using the Peanuts theme as a recessional.
At my wedding I had tradtional music, the Purcell as recessional and “Prince of Denmark’s March” for the processional. The latter is pretty good, and can be made to fit different lengths of time. There’s some short Bach cantatas that are also lovely, you might look into those.
In any event, let us know what you choose!
Getting married, huh?
I suggest Mars, Bringer of War.
I love Rimsky-Korsakov’s Procession of the Nobles from Mlada. It’s long, but could be used as processional for the entire wedding party or cut to a 1 minute portion.
From YouTube:
And definitely not overused.
Our processional was Scotland the brave played on bagpipes inside lol
Contemporary, but with a stately feel to it, and the lyrics are sweet: Tish Hinojosa’s “Who Showed You the Way to My Heart?”.
It was a very small outdoor wedding, sung by my new brother-in-law, and it fit perfectly.