In the introduction to her 1986 book, English Poetry of the Second World War: A Biobibliography, Catherine W. Reilly ranks the number of appearances various poets make in the 87 anthologies of World War II poetry she inventories:

Roy Fuller (25)
Alun Lewis (24)
Sidney Keyes (21)
Stephen Spender (19)
Keith Douglas (18)
John Pudney (18)
Alan Rook (17)
Louis MacNeice (15)
Henry Reed (15)
W.H. Auden (14)
G.S Fraser (14)
Dylan Thomas (14)
John Waller (14)
Emanuel Litvinoff (13)
Henry Treece (13)
Cecil Day Lewis (12)
Herbert Corby (11)
Nicolas Moore (11)

[p. xiii]

Reilly's bibliography doesn't dispel the idea that Henry Reed's Lessons of the War are the most-anthologized poems from the Second World War, just that Reed doesn't necessarily appear in the majority of anthologies of Second World War poetry.

The fifteen anthologies on Reilly's list that Reed appears in are:

An Anthology of Modern Verse, 1940-1960, chosen by Elizabeth Jennings (London: Methuen, 1961)

Components of the Scene: Stories, Poems, and Essays of the Second World War, edited by Ronald Blythe (Harmondsworth, Middlesex: Penguin, 1966)

English Poetry, Book 5: Modern Verse, edited by W.M. Smyth (London: Edward Arnold, 1971)

I Burn For England: An Anthology of the Poetry of World War II, selected by Charles Hamblet (London: Leslie Frewin, 1966)

The Martial Muse: Seven Centuries of War Poetry, edited by Alan Bold (London: Wheaton, 1976)

More Poems from the Forces: A Collection of Verses by Serving Members of the Navy, Army, and Air Force, edited by Keidrych Rhys (London: Routledge, 1943)

New Poems, 1944: An Anthology of American and British Verse, with a Selection of Poems from the Armed Forces, edited by Oscar Williams (New York: Howell, Soskin, 1944)

Poetry of the Forties, edited by Robin Skelton (Harmondsworth, Middlesex: Penguin, 1968)

Poetry of the 1940s: An Anthology, edited by Howard Sergeant (London: Longman, 1970)

The Poetry of War, 1939-45, edited by Ian Hamilton (London: Alan Ross, 1965)

The Terrible Rain: The War Poets 1939-1945, selected by Brian Gardner (London: Eyre Metheun, 1978)

These Years: An Anthology of Contemporary Poetry, edited by Howard Sergeant (Leeds: E.J. Arnold & Sons, 1950)

The Voice of Poetry, 1930-1950: An Anthology, edited by Hermann Peschmann (London: Evans Bros., 1950)

War Poetry: An Anthology, edited by D.L. Jones (London: Pergamon, 1968)

Where Steel Winds Blow, edited by Robert Cromie (New York: David McKay, 1968)

Also, someone needs to make poor Herbert Corby (Hampdens Going Over [1945]) a Wikipedia page.