Arrivals, The Shipping Gazette and Sydney General Trade List, 27 February 1860, p 34, http://nla.gov.au/nla.news-article161172311:

Wonga Wonga (s.), 200 tons, Captain Walker, from Melbourne 16th instant. Passengers—[...] Archdeacon McEncroe, [...], Rev. Messrs. Daly and Sealey, [...]

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Deaths, Sydney Morning Herald, 7 August 1860, p 1, http://nla.gov.au/nla.news-article13044017:

Of your charity, pray for the soul of the Rev. Father Bernard Daly, who departed this life on the 6th instant.

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Sydney Morning Herald, 7 August 1860, p 1, http://nla.gov.au/nla.news-article13044016:

A Dirge will be Chaunted to-morrow evening, at 7 o'clock, in St. Mary's Cathedral, and high Requiem Mass celebrated on Wednesday morning, at nine o'clock, for the repose of the soul of the late Rev. Father Bernard Daly. His funeral procession will move from the Cathedral immediately after Mass, at about 10 o'clock.

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Funeral Rites, Sydney Morning Herald, 9 August 1860, p 8, http://nla.gov.au/nla.news-article13044167:

Yesterday morning a mass for the dead was performed in St. Mary's Cathedral by the Very Rev. Dean O'Connell. The service was for the repose of the soul of the Rev. Bernard Daly, who died on Monday last at St. Vincent's Hospital, of lingering consumption. Nearly all the Catholic clergy of Sydney assisted in the ceremonies, which attracted a considerable congregation. The coffin containing the body was placed in the west chancel, and was covered with a purple pall, whilst lighted candles stood on both sides of the deceased's narrow cell. Shortly after the termination of the mass, the absolution was performed; the body was then removed from the Cathedral to the hearse outside. The Guild of Sts. Mary and Joseph, preceded by their office-bearers, led the way to the Catholic Cemetery, and were followed by the Rev. Dr. Gregory and Father Ford in the Archbishop's carriage. The hearse came next, and then the mourning coaches, cabs, and the spectators of the procession. The rev. gentleman, we are informed, had been but a few months in Australia, whither he voyaged in the hope of improving his health. He was too far gone, however, to receive any benefit from our climate, and was never well enough to officiate in the sanctuary. His age, as we gathered from the record on his coffin, was forty-three. There was to have been a high mass yesterday morning, but circumstances prevented it.

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Fr T.J. Linane, From Abel to Zundolovich, Vol. 2: Biographies of Priests on the Australian Scene up to 1900, 1980, p 122:

Daley, Bernard, Irish
Brought to Australia by Archdeacon John McEncroe
His epitaph well could be: I came here to die. In the Maynooth List of priests drawn up by Canon Hamill and published in the 'Irish Ecclesiastical Record' (ICR) there is an entry: No. 1645, ordained for Dublin in 1841. It would be a fair guess that the Maynooth-educated is identical with the priest that Archdeacon McEncroe took under his wing when returning after a somewhat hectic mission in 1859 — a mission which smoothed out to some degree the complaints of the clergy assembled at the historic Campbelltown Conference the year before. McEncroe left the sickly Bernard to the mercies of the Melbourne climate and friends. He stayed for three months, the T.B. condition growing worse from day to day. Whether he was on active duty in Sydney in the months that followed isn't clear. Anyhow the dread disease couldn't be controlled, and so Father Bernard died in St. Vincent's Hospital on 6 August 1860. When the Old Devonshire Cemetery was dismantled to make way for the railway age, the remains were transferred to Rookwood.
(Footprints, Vol. 3, No. 6)
Information: Rev. F. Bourke CM, Vincentian Historian.