Quoted from Scholes' God Save the Queen! (pp. 284-287):

             THE CLAIM FOR HENRY CAREY (c.1690-1743)

    This name has in the past been attached to the tune and poem of God save the King in many British song collections, and it is still often attached to the tune when this appears in American collections married to other words; moreover Histories of Music, and similar works on both sides of the Atlantic, have given Carey the credit for the song's composition. The latest edition of Carey's poems (by T.F. Wood, 1930) includes the poem of God save the King, and attempts to justify the inclusion. That poem was not, however, included in any collection of Carey's works published during his own lifetime--for instance, his Poems on Several Occasions (1713, 1720, and 1729, enlarged on each republication ), and his The Musical Century in One Hundred English Ballads (with music by himself; 1737, 1740, and 1743).
    Whence comes this widespread idea that Carey composed the words and music of God save the King? From his son, George Savile Carey, who in 1795 (i.e. more than half a century after his father's death) first put it forth, nothing having ever been heard of such a suggestion before that date. His object is not in doubt; he thought that by establishing a claim of this sort he might dig up a pension.
    Now G.S. Carey himself could have no possible memory of any of his father's doings; he never even saw his father, being a posthumous child. But he managed to throw together a flimsy body of evidence, based on hearsay, such as the following:

    I have heard the late Mr. Pearce Galliard, an able counsellor in the law, and a colleague of my father's, who died some time ago at Southampton, assert, time after time, that my father was the author of 'God save the King', and that it was produced in the year forty-five and six. (G. S. Carey, The Balnea, 1799.)

(Here G.S. Carey shows that he did not even know the date of his father's death, which, as above mentioned, was 1743.)
      Then he stated, with solemnity, the fact that in a collection of songs published in 1750: 'It precedes a song of my father's.' (But what does that prove? Every song in the collection except the last one of all had to precede some other song. It is a law of Nature!)
    The 'hearsay' above mentioned includes the statement that John Christopher Smith, Handel's old amanuensis, remembered Henry Carey coming to him 'with the words and music, desiring him to correct the bass'. This came via Smith's medical attendant at Bath, Dr. William Harrington, who added that he had again asked Smith, and that Smith had definitely confirmed the statement. He says, 'His advanced age and present infirmity render him incapable of writing, or desiring to be written to, but on his authority I pledge myself to its truth.'
    Now here is the testimony of an old man in an advanced state of decay (he died less than four months later) as to an incident alleged to have occurred over half a century before. The incident is not easily credible, since we can hardly believe that Carey, who had composed and published very many operas, cantatas, &c., could not effectively and correctly harmonize a simple melody like that of God save the King, which could be harmonized by any present-day musical schoolgirl who has had twelve months' instruction in harmony. Most likely Smith's memory had gone wrong: he was thinking not of Carey but of some other person who had brought to him some patriotic song to correct; or it may just be that Carey had discussed with him some other (and more elaborate) composition with loyal words, a composition that had now transformed itself in Smith's memory into God save the King.
    Carey published a number of patriotic compositions, and a similar suggestion of forgotten identity weakens the evidence of one or two people who, as has been alleged, told somebody or other that they had heard Henry Carey sing the song (as his own) half a century or more before. What they had presumably heard him sing was not this particular song but some other stirring patriotic effusion (Carey has one song, published in the Musical Miscellany, 1731, which actually ends with the words, 'God save the King and Queen': it may easily have been that which they heard).
    ...
    We do not know who composed God save the King, but we do know a number of musicians who did not and one of them is certainly Henry Carey....
    ...