Great that you've found your way to our community. Validating and verifying family lore, even if written down in books, can be a bit of a headache. The writer may have written down the story exactly as told, but the teller was perhaps not truthful, or may have forgotten details. Or the writer may have left out crucial details, or embellished the story.
One of the things that strikes me about the story is that the name of the ship isn't mentioned. A generic "man o war" from the British Navy. It is entirely possible Jeremiah was pressed into service of course, it was a very common practice. And the name may simply have been lost over time, or Jeremiah may have thought it was better to not mention it.
Two of the sources you can research further are the Ship's Musters and the Ship's Pay Books. These can be found in the National Archives in the UK
. A quick search I did on Jeremiah Donovan with a birth year of 1785 did not throw up anything, but that doesn't mean he wasn't there of course. Even musters weren't all that accurate, although Pay Books were supposed to be at least a bit more accurate as they were a register of (part of) the ship's finances. Sometimes, if you're lucky enough to find an ancestor in the ship's musters, it will have some wonderful details about the way they looked (a physical description), others only mention name and maybe a place and date of birth.
A quick search through the Cork records
doesn't throw up an exact Jeremiah Donovan in 1785, but of course that can be off by a few years here or there. I'd advise you to do a broad sweep with 5 or even 10 years on either side of 1785 and pay special attention to those without death dates.
I hope these two sources at least give you somewhere to look further and another angle may be the British Navy ships that were docked in Sydney, maybe 15-20 years after his birth date, and any local historic police registers. Jumping ship was not without risk and you could become a wanted man by doing so!
Hopefully some of our experts will also be able to jump in with some tips as to where you could expand your search to.