"Once in Royal David's City" is very Anglican, otherwise no call to have heard of it. For that matter, so is "In the Bleak Midwinter".
"In the Bleak Midwinter" is a poem by Christina Rosetti (the painter Dante Gabriel Rosetti's sister):
In the bleak midwinter
Frosty wind made moan,
Earth as hard as iron,
Water like a stone;
Snow had fallen,
Snow on snow,
Snow on snow,
In the bleak midwinter,
Long ago.
Our God, heaven cannot hold him,
Nor earth sustain;
Heaven and earth shall flee away
When he comes to reign;
In the bleak midwinter
A stable place sufficed
The Lord God incarnate,
Jesus Christ.
Enough for him, whom Cherubim
Worship night and day
A breast full of milk
And a manger full of hay.
Enough for him, whom angels
Fall down before,
The ox and ass and camel
which adore.
Angels and archangels
May have gathered there,
Cherubim and seraphim
Thronged the air;
But his mother only,
In her maiden bliss,
Worshipped the Beloved
With a kiss.
What can I give him,
Poor as I am?
If I were a shepherd
I would bring a lamb,
If I were a wise man
I would do my part,
Yet what I can I give Him
Give my heart.
The best known setting is by Gustav Holst, but there is also a beautiful one by Harold Darke.
"Once in Royal David's City" was written by Cecil Frances Alexander, who also wrote "All Things Bright and Beautiful" and a bunch of other poems and hymns. It's very long, and all the verses aren't usually sung.
Once in royal David's city
Stood a lowly cattle shed,
Where a mother laid her baby
In a manger for His bed:
Mary was that mother mild,
Jesus Christ her little child.
He came down to earth from heaven,
Who is God and Lord of all,
And His shelter was a stable,
And His cradle was a stall;
With the poor, and mean, and lowly,
Lived on earth our Savior Holy.
And through all His wondrous childhood
He would honor and obey,
Love and watch the lowly Maiden,
In whose gentle arms He lay:
Christian children all must be
Mild, obedient, good as He.
For He is our childhood's pattern;
Day by day, like us He grew;
He was little, weak and helpless,
Tears and smiles like us He knew;
And He feeleth for our sadness,
And He shareth in our gladness.
And our eyes at last shall see Him,
Through His own redeeming love;
For that Child so dear and gentle
Is our Lord in heaven above,
And He leads His children on
To the place where He is gone.
Not in that poor lowly stable,
With the oxen standing by,
We shall see Him; but in heaven,
Set at God's right hand on high;
Where like stars His children crowned
All in white shall wait around.
http://www.youtube.com/watch?v=S01sjiXzpLk
Lichfield, btw, was Dr. Samuel Johnson's home town.