I can confirm from my own offline resources:
Oct 1802
John DWYER aged 44, tried at CLONMEL, sentenced to Life, transported per Atlas 2, arrived 30 October 1802. Conditional Pardon # 96 (and this info potentially conflicts with other info later in this post)
Oct 1802
John DWYER aged 28, Convicted LIMERICK, sentenced to Life, Transported per Atlas 2, arrived 30 Octobrer 1802, Absolute Pardon #327
August 1806
John DWER per Atlas, a prisoner, at the Battery, Sydney (note spelling on the surname is as per my resource)
1810
Letter from John DWYER, ex Ship Atlas 1802, seeking mitigation of his sentence
29 Feb 1812, Conditional Pardon for John DWYER shows he was tried at Limerick, received a Life sentence, and had arrived by the ship Atlas in 1802. (And this info potentially conflicts with earlier info in this post)
31 Jan 1818, Free Pardon for John DWYER, tried Limerick, sentenced Life, ship of arrival Atlas 1802
1814
30 June 1814 letter, John DWYER ex Atlas 1802, Paid from the Police Fund for erecting for the public use the stone Reservoir and Pump in Bent Street (Sydney CBD)
July 1822 letter, John DWYER ex Atlas 1802, receiving an assigned convict
Sept 1822
John DWYER at Liverpool, 80 acres, cleared 30 acres, 20 acres under wheat, 7 acres under maize, 1 acre potatoes, 9 bushels wheat, 60 bushels maize, 1 horse, 15 horned cattle 20 hogs.
SEPT 1822 MUSTER John DWYER, Life, arrived per Atlas 2, “AP” (Absolute Pardon) Residence APPIN
26 Dec 1822, John DWYER, Atlas 1802, Appin NSW, Convicts ex the ship Countess of Harcourt assigned to John Dwyer at Liverpool
April-May 1823, letter John DWYER Atlas 1802, District Constable
28 April 1823, John DWYER, Atlas 1802, appointed Chief Constable Appin district,
7 May 1823, John DWYER, Atlas 1802, Appointed Chief Constable and Poundkeeper district of Appin
1 Oct 1823, John DWYER, Atlas 18102, On the return of the Police Establishment in the Districts of Appin and Illawarra as Chief Constable and Poundkeeper Appin
15 Nov 1823, letter John DWYER ex Atlas 1802, resigned due to ill health
Jan, Feb, Oct 1824
Letter re Convict assigned to John DWYER ex Atlas 1802 in the District of Appin
Letter re Convict returned to Government from John DWYER is Atlas 1802 in the District of Appin
30 March 1824 Letter John DWYER, Atlas 1802, letter of Resignation
30 August 1824 Letter John DWYER, Atlas 1802, letter Memorial (Petition)
23 June 1825, Letter John DWYER, Atlas 1802, On list of persons who had orders for Grants of Land
23 June 1825 Letter John DWYER Atlas 1802 Lands Granted/Reserved by Sir Thomas BRISBANE as governor of NSW.
Perhaps the two chaps were different denominations, but even that will not help to separate the records, as St Philips C of E, Sydney's parish records contain many baptisms for all denominations in the 1810-1821 era when Lachlan Macquarie was Governor. (As per his general orders issued 1810 etc about recording of baptism and burials).
Cheers, JM