The American Civil War produced a whole slew of songs. This is one of the best known, though I prefer the Confederate parody:
Just before the battle, Mother,
I was drinking mountain dew.*
And when we saw the Rebs, dear Mother,
to the rear I quickly flew!
Farewell, Mother, for you’ll never
see my name among the slain,
for he who runs away, dear Mother,
lives to fight another day!
*Moonshine, not soda pop. 
That’s very nice, but in terms of “unofficial national songs”, it can’t beat Katusha:
Or Kalinka:
Kids of my generation (and before) might remember this one from singing around the campfire at summer camp or as Scouts, but I don’t know how many would recognize it today:
For Russia, I would think Stenka Razin would come high on the list:
It was repurposed by the Seekers as The Carnival is Over:
Speaking of The Seekers:
When it comes to “national” songs of Canada, it’s hard to beat Gordon Lightfoot:
I’m not even British, and “Jerusalem” is first song I thought of when I saw the thread title!
Land of Hope and Glory sprung to my mind (for the English).
Actually, Jerusalem’s the one that mentions England specifically (though it could be generalised with just a minor adjustment of the words). Land of Hope and Glory is just generally imperial.
I like Jerusalem - I’m not religious, and, despite mentioning the word ENG-ER-LAND, it actually isn’t very nationalistic. More a desire to create a better society.
As a Canadian, I’ve never heard that G Lightfoot song, so I don’t think it’s the one!
I was thinking more of this one,
Or this one,
You missed out the national anthem of Wales, Hen Wlad Fy Nhadau. Best sung by a large rugby crowd.
Sticking with a Welsh theme (my wife is Welsh so have been subjected to many of these, and the Welsh do love a sing song), there’s Sosban Fach, which they like to sing VERY loudly at parties while banging the tables to the beat of the chorus…
And my personal favourite, International Velvet, by Catatonia, which has the immortal chorus ‘everyday, when I wake up, I thank the Lord I’m Welsh’
Oh, forgot one. Tom Jones, green Green Grass of Home.
I wonder how many people realize this anthem’s melody is actually the bridge in the middle of “Jupiter, the Bringer of Jollity” in Holst’s The Planets:
And not this one? 
It’s not unusual.
I know one verse, but I cannot sing it with the proper operatic aplomb:
To thee belongs the rural reign,
thy cities shall with commerce shine,
and lands far o’er the spreading main
shall stretch a hand to grasp with thine!
Phew! It’s exhausting to just type it! 
I have been ruined by ren fairs. Every time I hear this, I now think of Scotland’s Depraved
I have been ruined by the videogame Mafia II.
Same here (I spent my summer weekends in the '90s working at our local fair). “Bring me a sheep, for I am lonely tonight.”