Well, let’s see…
“O Canada” by Calixa Lavallee
“Northwest Passage”, “The Nancy” and “The Bluenose” by Stan Rogers
The Canadian lyrics to “This Land is Your Land” (“From Bonavista to Vancouver Island/From the Arctic Circle to the Great Lake waters…”)
Oh, and “The War of 1812 (The White House Burned, Burned, Burned, and We’re The Ones Who Did It)” by the Arrogant Worms. 
Americans, be thankful you have The Battle Hymn of the Republic, and America the Beautiful. Those two (especially the former) are amazing pieces of music.
As for my native Australia, well of course I know Advance Australia Fair (including the lesser known imperialist British verses), and the mighty Waltzing Matilda, but I’ll post here the lyrics to the more whimsical Home Among the Gum Trees:
1 I’ve been around the world
A couple of times, or maybe more,
I’ve seen the sights, I’ve had delights
On ev’ry foreign shore,
But when my friends all ask me
The place that I adore,
I tell them right away.
CHORUS
Give me a home among the gum trees,
With lots of plum trees,
A sheep or two, a kangaroo.
A clothesline out the back,
Verandah out the front
And an old rocking chair.
2 You can see me in the kitchen
Cooking up a roast,
Or vegemite on toast,
Just you and me, a cup of tea.
Later on, we’ll settle down,
And mull up on the porch
And watch the possums play.
CHORUS
3. There’s a Safeway on the corner
And a Woolworths down the street,
And a New World’s just been opened
Where they regulate the heat,
But I’d trade them all tomorrow
For the simple bush retreat
Where the kookaburras call.
CHORUS
4 Some people like their houses
With fences all around,
Others live in mansions,
And some beneath the ground,
But me, I like the bush, you know,
With rabbits running round
And a pumpkin vine out the back.
CHORUS.
CODA
And an old rocking chair.
That line always makes me laugh TLD. I think we don’t really have the variety of patriotic songs that Americans are blessed with, and I say that without a hint of irony. Apart from the ones you mentioned, I might include And the band played Waltzing Matilda as a song that means a lot to many Aussies, even though The Pogues recorded the best version, IMO anyway.
There was a classic episode of Play School where they sang Home Among the Gum Trees for the kiddies, and when John Hamblin came to “Later on, we’ll settle down, and mull up on the porch”, he barely held it together. The co-host was having trouble stifling her smile too. It was brilliant.
I knew it was coming, and expected them to change that line to something else. Cool of then not to though.
Yes, John and Noni made a wonderful pair, especially when they were romping around the studio pretending to be dogs woof!
What about ‘True Blue’ by that moron John Williamson?? While his ‘ocker’ image bugs me, the song always brings a lump to my throat.
My Country 'Tis of Thee, This Land is Your Land (sort of), a few sections of The Star Spangled Banner, America the Beautiful, Grand Old Flag, the Taiwanese Flag Song, Taiwanese National Anthem, Waltzing Matilda, and… I think that’s it. Not counting parodies and such…
Phew! 
How could I forget Noni Hazelhurst’s name? Musta been a brainfart. She was great on that show. They both were. John Hamblin just had the right face for it. I feel sorry for the poor bugger whenever he goes into a pub though: “Hey, it’s that Play School bloke! Hel-lo! I’m drink-ing BEER to-day!”
And you’re right, John Williamson is a major pain in the arse. His politics are pretty iffy too. But I must admit, I do like True Blue, and also Raining On the Rock.
Norrybaby, And the Band Played Waltzing Matilda is truly moving.
Tons of 'em. I went to public school during the Reagan years, and music class consisted almost entirely of patriotic schlock (with the occasional musical soundtrack thrown in). Shoot, I know patriotic songs which have probably never been heard outside of school corridors:
In the days of '76, my boys
We ever must recall
Our fathers took their muskets then
For freedom to stand tall
Upon the plains of Lexington
They caused the foe to fall …
I swear, I think my music teacher made that one up.