Alexander Taylor was appointed early in 1803 as Assistant-Surgeon to the Euryales, frigate.
He was appointed Surgeon in the Royal Navy on 23 July 1804 and served on the Railleur in that year.
In January 1807 he was appointed to the Inconstant, frigate, flag-ship at Guernsey. He served on board the Thalia, frigate, at the blockade of Cherbourg, on the expedition to Flushing, and in the West Indies. In 1810 was appointed to the Pompee, 74, and afterwards to the Havannah, frigate, when he served on the Coast of France at the blockade of the French fleet at Toulon, in the Adriatic at the capture of the fortress at Zara, and at a variety of captures. After three years' severe service he was appointed to the Milford, 74, flag-ship of Admiral Fremantle, at Trieste, where he remained until the peace in 1815. 
Alexander Taylor was employed as Surgeon Superintendent on two convict ship voyages to Australia:
In the Navy List of 1842, Alexander Taylor (R.N. 23 July 1804) is noted as having a pension of £40 12s 4d per annum as retired surgeon of the convict ship Surprise. (This was the prison hulk Surprise which was moored at the Cove of Cork 1822 - 1837)
According to the House of Commons Papers 1826 a Surgeon employed full time on the hulks would receive a salary of £80 per annum.
Alexander Taylor died at Upper Park, Queenstown, Ireland on 7 September 1850. It was noted that he had spent forty-six years as Surgeon in the Royal Navy.
Notes and Links
1). Marriage - 1st July at Cove, Captain Thomas Hilton, 19th Regiment, to Annie second daughter of Alexander Taylor Surgeon R.N., 
2). Marriage - At Monkstown Church, Captain Robert Campbell 73rd regiment to Dora, youngest daughter of Alexander Taylor Esq. surgeon R.N. of Monkstown Cork. 
3). File of papers relating to appointments to posts onboard the new convict ship hulk 'Surprize', Cork - includes draft or copy letter signed by William Gregory, Under Secretary, Dublin Castle, to Samuel Hollingsworth, inspector of convict hulk 'Surprize', reporting on the appointment of Reverend Philip Panter as Protestant chaplain; Reverend Thomas Croke as Catholic chaplain; Alexander Taylor as surgeon; and Mr Kavenagh as ship keeper to the hulk, July 1823.