This little gem has apparently only come to any attention once; in 1909. (glancing through the modern books on Sir Sidney, found no mention of midshipman John Tillery.)

In 2003, Christie's auction called it; AN INTERESTING AND HISTORICALLY IMPORTANT JOURNAL KEPT BY JOHN TILLERY, MIDSHIPMAN OF H.M.S TIGRE (COMMANDED BY SIR WILLIAM SIDNEY SMITH), CAPTURED AT THE SIEGE OF ACRE AND A PRISONER-OF-WAR BETWEEN 21ST MARCH 1799 AND 13TH FEBRUARY 1800 - Went for almost $5,000 http://www.christies.com/lotfinder/Lot/an-interesting-and-historically-important-journal-kept-4097709-details.aspx

Ended up here http://collections.rmg.co.uk/archive/objects/262970.html

As far as I could find, extracts from Tillery's journal can only be found in The United Service magazine: v.38 1908/09. (p.201)

It's worth quoting the opening:

A CAPTIVE OF BONAPARTE’S IN SYRIA.
BEING EXTRACTS FROM THE JOURNAL OF JOHN TILLERY OF
VIRGINIA, SERVING ON BOARD H.M.S. TIGRE, COMMANDER
SIR WILLIAM SIDNEY SMITH, AND TAKEN PRISONER BY
THE FRENCH.
BY FRANK WARD.

THE personal narrations of even humble participators in great movements, possess a fascination that the polished and exact record of the historian usually lacks, often bringing to light interesting and hitherto unknown details of such events, and the actors therein. The Journal of John Tillery, sometime Gunner’s Mate of H.M.S. Tigre, covering a period of nine months’ captivity in Syria and Egypt, affords us a few such glimpses of Bonaparte’s disastrous campaign in Egypt. During that troublous time few actions equalled, and none surpassed, the gallant defence of Acre by Captain (afterwards Admiral) Sir William Sidney Smith. For a period of sixty days, by day and night, he successfully resisted attacks from within and without, assassination and plague, fire and sword ; having the ultimate satisfaction of seeing his great adversary retire, beaten and discredited—his hopes of Eastern dominion finally shattered.

Bonaparte, who never afterwards ceased to inveigh against“ this miserable little hole which came between him and his destiny,” once remarked to an intimate that, “If St. Jean d‘Acre had fallen, he would have changed the face of the world, and been Emperor of the East.” It is with the siege of Acre that John Tillery’s Journal opens.

https://babel.hathitrust.org/cgi/pt?id=coo.31924066186572;view=2up;seq=218