An additional quotation from Percy A. Scholes, ed., The Oxford Companion to Music, 10th ed. (1970, p. 411):

10. Various Untenable or Doubtful Claims. Claims that have been made for the composition of this tune by Lully and Handel and others (or even for its final arrangement by them) are not worth entering into here, as under examination they have been found to be vitiated by gross errors of date, etc. The claim on behalf of Henry Carey is equally untenable;.... The suggestion has been made that the first printed version ... is the work of James Oswald (q.v.), who may have been the editor of Thesaurus Musicus; this cannot at present be proved or disproved.

Scholes had researched extensively and written a book on this song: God Save the Queen!: The History and Romance of The World's First National Anthem (Oxford University Press, 1954; xviii+328 pp., with 25 plates).
      James J. Fuld, The Book of World-Famous Music, 5th ed. (Dover, 2000, pp. 249-50) wrote:

Chappell, p. 704, stated that the anthem was first printed in Harmonia Anglicana; inasmuch as no copy of such work had been found containing the anthem, it had generally been agreed that Chappell erred. However, a unique copy of Harmonia Anglicana has recently been found at the Library of Congress, and there is no question but that this volume precedes the volume retitled Thesaurus Musicus, previously believed to contain the first printing of the anthem. Harmonia Anglicana, "A Collection of ... Songs, Several of them never before Printed" (London), was apparently published in one volume, the anthem appearing on p. 22 in two stanzas only. A publication date between April 20 and Nov. 16, 1744, is probable. (See Plate II.)