The Belfast Newsletter mentioned John Connellan in July 1805 - Conlan, the apothecary from Dundalk, who had been an approver in 1798, and has been ever since supported by the bounty of Government, has, volunteered to go to Botany Bay with the convicts now in the river. He has, it is said, lately qualified himself to act as surgeon, and being unwilling to return to his native country, has received an appointment in the colony. 
John Connellan was employed as surgeon on the convict ship Tellicherry
in 1806 -
Secretary A. Marsden to Governor King.
31st July, 1805.
The Lord Lieutenant of Ireland, having appointed Mr. John Connellan (who has been recommended to His Excellency by the Medical Board) to be Assistant Surgeon on board the Tellicherry Convict Ship, on her Voyage from Ireland to New South Wales; I beg leave to recommend him to Your Notice, And to request You will assist him in obtaining a Conveyance for his return to Europe, in Case he shall not determine to remain abroad. Should Mr. Connellan have occasion for money to defray his Expenses, His Excellency requests You will (should you approve of his Conduct) Advance him any Sum on Account of his Salary not exceeding: Two hundred pounds Irish Money, And Your Bills for the Same shall be duly honored.
Secretary A. Marsden to Governor King.
17th August, 1805.
A Mr. John Connellan has been embarked in the Tellicherry as Assistant Surgeon in the Ship, And he has some intention of Settling in New South Wales Should he be disposed to do so on his Arrival; it will give the Lord Lieutenant Satisfaction, if attention be paid to him by You and any Service rendered to him - He has been useful to Government in a Way that he will explain to You, And his general Conduct for some Years past, in which I have had intercourse with him, has been Such as makes me think him deserving of Your Notice. 
arrived in Port Jackson on 15th February 1806. John Connellan was appointed to act as surgeon at Norfolk Island
soon afterwards. D'arcy Wentworth who had been stationed at Norfolk Island, returned to Sydney -
Governor King to Earl Camden,
Sydney 22 February 1806 (Extract)
Being informed that Mr. James Thompson
, assistant surgeon now on leave in England, has no intention of returning to this colony, and not knowing what the event of the sentence passed on Mr. Savage, another of the assistant surgeons, may be, and there being a great want of the necessary medical gentlemen to do duty here, I have taken it upon me to appoint Mr. John Connellan, who has been so strongly recommended by the Irish Government, as stated in the enclosed copy of Mr. Secretary Marsden's letter, to act as surgeon at Norfolk Island until your Lordship's further commands are received thereon
Governor King to Secretary Marsden
22 February 1806 -
Respecting Mr. Connellan, I shall be happy to render every service in my power; but as to placing him on the establishment of this colony, the numbers of medical gentlemen are provided for in England by the Parliamentary estimate, which I am commanded by H.M. Government not to exceed, as their commissions are signed by the King. Whether there is an actual vacancy or not in that department I cannot tell. Owing to some of the gentlemen being in England, and, as I hear, not intending to return, this may eventually occasion some vacancy, the chance of obtaining which I have offered to Mr. C. dependant on the approbation or rejection of H.M. Government, which he has accepted and is appointed to act as surgeon of Norfolk Island until I receive directions on that head; and any interference of yours with this arrangement on that behalf will be instrumental to that gentleman's success.
I have also directed to Commiss'y-General to draw on you for the sum of two hundred pounds Irish, for the use of Mr. Connellan, whose receipts and voucher I have the honour to enclose. -
Notes and Links
Return of the number of inquests held in Ireland in 1841, House of Commons Co. Louth
?same John Connellan
 Belfast Newsletter 19 July 1805
 HRA Series 1, Vol. V.p. 550
 HR NSW., Vol. VI 22 February 1806, pp.20-21