These are the earliest citations I could find for the phrase "loss of generality" in JSTOR. Note how they all slightly differ from the strict "without loss of generality" form. Also note how they're all from William R. Hamilton.

... and many of the new partial differential coefficients vanish, without producing, by this simplification, any real loss of generality

• Third Supplement to an Essay on the Theory of Systems of Rays, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 17, (1831), pp. v-x, 1-144

Mr. Jerrard has therefore accomplished a very remarkable simplification of this general problem, since he has reduced it to the problem of discovering two real functions of two arbitrary real quantities, by showing that, without any real loss of generality, it is permitted to suppose ...

• On the Argument of Abel, Respecting the Impossibility of Expressing a Root of Any General Equation above the Fourth Degree, by Any Finite Combination of Radicals and Rational Functions, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 18, (1839), pp. 171-259

And if we farther simplify the formulae by supposing $ a = 1, b=0, c=0, d=0$, which will be found in the applications to involve no essential loss of generality ...

• Researches Respecting Quaternions. First Series, William Rowan Hamilton, The Transactions of the Royal Irish Academy, Vol. 21, (1846), pp. 199-296

These are the earliest citations I could find for the phrase "loss of generality" in JSTOR. Note how they all slightly differ from the strict "without loss of generality" form. Also note how they're all from William R. Hamilton.

... and many of the new partial differential coefficients vanish, without producing, by this simplification, any real loss of generality

• Third Supplement to an Essay on the Theory of Systems of Rays, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 17, (1831), pp. v-x, 1-144

Mr. Jerrard has therefore accomplished a very remarkable simplification of this general problem, since he has reduced it to the problem of discovering two real functions of two arbitrary real quantities, by showing that, without any real loss of generality, it is permitted to suppose ...

• On the Argument of Abel, Respecting the Impossibility of Expressing a Root of Any General Equation above the Fourth Degree, by Any Finite Combination of Radicals and Rational Functions, William R. Hamilton, The Transactions of the Royal Irish Academy, Vol. 18, (1839), pp. 171-259

And if we farther simplify the formulae by supposing $ a = 1, b=0, c=0, d=0$, which will be found in the applications to involve no essential loss of generality ...

• Researches Respecting Quaternions. First Series, William Rowan Hamilton, The Transactions of the Royal Irish Academy, Vol. 21, (1846), pp. 199-296

EDITED TO ADD:

Here are the dates and authors for the first 20 instances of "loss of generality" in the above JSTOR search:

 1831   Hamilton
 1839   Hamilton
 1846   Hamilton
 1848   Stokes
 1854   Cayley
 1855   Cayley
 1856   Thomson
 1857   Cayley
 1860   Donkin
 1862   Cayley
 1863   Schlafli, as communicated (translated?) by Cayley
 1864   Cayley
 1866   Sylvester
 1867   Cayley
 1867   Cayley
 1868   Cayley
 1870   Strutt
 1871   Russell
 1873   Williamson
 1874   Cayley

While the three earliest citations are due to Hamilton, fully half of the first twenty instances in the JSTOR database are due to Cayley. Of course, the JSTOR database is not comprehensive; in particular, it does not include The Transactions of the Cambridge Philosophical Society which contains the earlier Stokes citation that Brendan McKay found.