Cornelius Kelly was entered in the Navy List of Medical Officers in 1814.
Surgeon Superintendent Woodman 1826
Cornelius Kelly took over the duties of Surgeon Superintendent on the convict ship Woodman at the Cape of Good Hope when the surgeon of the ship John Rodmell died during the voyage.
The Colonial Times and Tasmanian Advertiser reported the arrival of the Woodman in Tasmania in May 1826 - On Monday, arrived the ship Woodman, Capt. Leary, from England with 150 male prisoners. On board this ship have arrived the Rev. Mr. Mummery, (a regular clergyman), Mr. Christmas, the Bank Clerk, and *Mr. Jefferson (or some such name), a Swede, all of whose cases excited much observations in England. 
(* this was Jorgen Jorgensen)
The Woodman arrived in Sydney later in May - 'The Woodman, Capt. Daniel Leary, burthen 419 tons, from Sheerness, 6th December, Cape of Good Hope, 4th March with 146 male prisoners, 4 having died on the passage, as well as the Surgeon Superintendent Mr. Rodmell. Mr. Kellie of H.M. Ship Helicon, undertook the charge at the Cape. The guard consisted of Captain Wakefield, and Ensign Innes of the 30th and 2 sergeants and 7 rank and file of the 57th.' 
Free passenger Phillip Lander Fell arrived in Sydney on the Woodman
The following articles were published in the newspapers in 1826. Cornelius Kelly wasn't mentioned, however apparently assumed innuendo was aimed at him: -
We understand from a Gentleman of great respectability, that the crew of the ship Woodman, went on shore at the Cape de Verde Islands, and killed some cattle belonging to the settlers there, and that the inhabitants of the island were themselves maltreated. We shall be happy to see this contradicted by our Contemporaries. - The Monitor 7 July 1826.
A malicious wanton and unfounded assertion relative to the crew of the ship Woodman, having made its appearance in the Monitor of yesterday, Mr. Leary, commander of that ship (and on whom such unprincipled attack evidently appears to have been made) has no hesitation in denying the truth of the statement, from whatever respectable quarter it may have emanated. At the Island of St. Vincent (Cape de Verds) w watering party, consisting of a few men of the guard and crew, did, with the conduct and sanction of the natives, shoot some wild cattle with which the mountains abound; and, so far from the inhabitants having been maltreated, Mr. Leary (who was not himself out of the ship, on any occasion, during the passage from England to Van Diemen's Land) was informed by the commanding officer of the guard, that he was received by, and took his leave of, the Governor on the most friendly terms. - Sydney Gazette 8 July 1826
We are requested by Dr. Kelly to state, that the reports which have been so industriously propagated by the Monitor, which have excited so much interest from their monstrosity, did not originate with him, as he had not at that time joined the ship. - Sydney Gazette, 22 July 1826
Back in England in 1828 Cornelius Kelly lodged at the house of Mr. and Mrs Mercer at Somers-town. He was a witness at the inquest into the death of Mrs. Prudentia Mercer. He had little to do with the case, however managed to disrupt the proceedings:
(Extract) Mrs. Mary James resides opposite the house of Mr. Mercer, who is a chemist and druggist. About half past 12 o'clock on Monday night a gentleman named Kelly came to her house, and said that Mrs. Mercer had poisoned herself.
The individual mentioned by Mrs. James here stood up, and addressing the coroner, said, 'Sir, I will not be called gentleman; I will not have my name played with in such a manner. My name is Cornelius Kelly, and I am a surgeon in his Majesty's royal navy.'
The coroner said, he conceived the way he and the witness had described Mr. Kelly to be in a most respectful manner; and if he did not like the designation of gentleman, he could have no objection to strike it out in the deposition.
Mr. Kelly said - I am a surgeon in his Majesty's navy, and an Irishman, and a papist, and all that.
The coroner here interfered and said, that unless Mr. Kelly refrained from indulging in such conduct, he should be under the necessity of ordering the room to be cleared of strangers.
Mr. Kelly still persisted in demanding the word gentleman being struck out, and Cornelius Kelly, surgeon in his Majesty's royal navy being inserted instead.
Coroner - Well sir, you shall be accommodated. Mr. Sterling then erased the word 'gentleman' and introduced the designation by Mr. Kelly, and the inquest proceeded..........
The testimony of Mrs. Mary Harrison.....
Juror: Did you have any conversation with Mrs. Mercer? Yes, sir, respecting a servant girl that was going to leave her situation in consequence of a gentleman who lodged in the house having taken liberties with her.
Coroner: Who was that gentleman, was it Mr. Mercer? - No sir.
Juror: Who was it?.... Must I tell?
Juror: Yes, to be sure? Then it was Mr. Kelly, the gentleman who sits there
Mr. Kelly - It was I, sure enough, Cornelius Kelly, surgeon in his Majesty's royal navy. I only put my arm round her neck and gave her a kiss, as any other gentleman would do who had been taking a glass of grog. I declare to God I never spoke to the crature (creature) in private in all my life.
Mr. Sterling - Nobody said you had; there was no great harm in kissing the girl
Mr. Kelly - None in the wide world.
Cornelius Kelly died prior to 1843 when his youngest son passed away.......(On the 15th instant, at St. Colum's Court, Patrick James, youngest son of the late Cornelius Kelly, Esq., surgeon R.N. of Londonderry. - The Freeman's Journal 20 January 1843)
 Colonial Times and Tasmanian Advertiser (Hobart, Tas. : 1825 - 1827) Fri 5 May 1826 Page 2
 The Sydney Gazette and New South Wales Advertiser (NSW : 1803 - 1842) Wed 24 May 1826 Page 2
 The Standard 10 July 1828)
 The Sydney Herald (NSW : 1831 - 1842) Thu 14 May 1835 Page 4