See Maynard Lee Daggy's review of
Enfield in THE QUARTERLY JOURNAL CF
SPEECH, Volume 18, pages 123–129
(February 1932).

The edition reviewed by Daggy carries the imprint: "London: Printed for J. Johnson, St. Paul's Churchyard; by whom the only genuine and complete Edition is published. — (1744) 1799."

In Daggy's reproduction of the short dedication (found on pages a 2 and v below) in this 1799 edition two lines appear below the name William Enfield, as follows:

Warrington Academy
October 1, 1774

This presumably would fix the date of the first edition.

LERARIES
SPEAKER ; on
~~
Miscellaneous PIECES, -

AN ESSAY ON 292
ELOCUTION. e
By WILLIAM ENFIELD, LL.D.

LECTUR L R OF THE BE L L E S L E TT RES IN THE ACADEMY
AT WAR R IN GTON, -