I can find nothing of where he came from.

But the following indicates that a different spelling was used in his lifetime, from the Carte Calendar, vol 39, Jan to May 1664, at the Bodlean:

Petition of Luke Reilly [in MS.: "Reily"] to the Duke of Ormond

Date: [circa 5 February] 1664

Shelfmark: MS. Carte 159, fol(s). 210

Document type: Copy

Recites the circumstances under which petitioner was found guilty of "homicide by misadventure", for the killing of John Hamilton, one of the servants of William Baily, Bishop of Clonfert, in whose house petitioner was maintained by the Bishop's charity.

Prays for the grant of his Majesty's pardon.
_______________________________________________

There is also a list of all the bishops of Clonfert where he is given as William  Bayly also Bailie and said to have died in 1665 - shortly after the above plea.
_______________________________________________

He is not to be found in the Oxford or Cambridge alumni.  But he might have got his doctorate from Trinity College Dublin or from one of the Scottish universities.
_______________________________________________

Just as I was about to give up, I found this in the history of the town of Bailieborough at http://www.bailieborough.com/history/:

“Bailieborough A Plantation Market Town

“Bailieborough, like its neighbouring town Virginia and the other American Virginia has its origins in the plantation of Ulster. William Bailie, a Scottish "undertaker" or Planter, was granted the lands of Tonergie (Tandragee) in East Breffnie, by James I of England. He built a castle and enclosed the demesne by 1629. On his death the estate passed to his son, William, Bishop of Clonfert. The Bishop's daughter married James Hamilton and on the Bishop's death they inherited Bailieborough Castle. Henry Hamilton, their son, was a member of the IRA [who?], but died at the Siege of Limerick in 1689.”

Very useful!

And from the same site there is also:

“No Mean Village
 
“By Liam O' Ceallaigh

"Bailieborough is a very mean village in the same barony" (Clonkee).
Sir Charles Coote in his Statistical Survey of Co. Cavan prepared for the Royal Dublin Society, 1801.

"I know of no town more neglected or which has better capabilities than Bailyborough." (lbid.)

”The story of Bailieborough goes back to the early years of the 17th century. In 1610 William Bailie, a native of Ayrshire, was given a grant of 1000 acres in the proportion of Toneregie, now Tandragee, in the Barony of Clankee in Co. Cavan. Under the terms of the grant he was required to enclose a demesne of 350 acres. On this he was to build a bawn and within the bawn to erect a strong house or castle. He was also required to settle upon his estate a number of families of English or Scottish extraction. He was further required to establish fairs and markets, and also to establish courts for the administration of the law etc.

”In the Pynnar (Survey of 1619) we are told that William Bailie had taken possession of his lands in Cavan and that his castle was in the course of erection. It was also reported that a number of Scottish families had been settled on the estate. At a commission held in Castle Aubigny (Shercock) in 1629 to enquire into the progress being made by Bailie and the other grantees in the area in carrying out of the conditions set out in the terms of their several grants it was found that William Bailie had his castle completed and was living therein together with his wife and family, and that 28 British families had been settled on his estate.

”William Bailie had two sons, William and Robert. William was educated in Trinity College, Dublin, and was later ordained a minister.  [But Alumni Oxoniensis says he, as ‘William Baylie’, was educated at Glasgow and then got DD in Oxford in 1644.]   He served as rector in a number of parishes in Co. Cavan. In 1644 he received the degree of D.D. and two years later he was made bishop of Clonfert. His brother, Robert, entered the army. In 1640 he was reported as having command of a troop of Scottish soldiers in Cavan.

”During the rising of 1641 Bailie's Castle was attacked and captured by a body of Irish soldiers under Colonel Hugh O'Reilly. They held the castle and its inmates for a month and then departed carrying off a large number of cattle and horses.

William Bailie, senior, died about 1648 and his son, William inherited the castle and estate. The bishop had one daughter, Anne, who married James Hamilton, third son of John Hamilton of Coroneary Castle, and on the bishop's death in 1666, Bailieborough Castle and estate passed into the hands of the Hamiltons. James Hamilton's son Henry, succeeded his father. He was M.P. for Cavan, and during the Jacobite war he took the side of King William and was killed at the siege of Limerick. His successor was his son, another James Hamilton, of whom more later.

“During the years that the Bailies lived in Bailieborough Castle, a small hamlet or village grew up in Lower Drumbannon, near where the Castle River emerges from the Castle Lake. The houses were likely built of timber or mudwall and roofed with thatch. Later in the century, the Hamiltons demolished the village and had it rebuilt in Upper Drumbannon, overlooking the Town Lake. It is likely that this was the village that Sir Charles Coote wrote of in his survey in 1801.

”In 1720, James Hamilton was granted a charter for the holding of fairs and markets on stated dates "in Newtown, alias Bailieborough" but he seems to have had a change of mind, for in 1724, he sold his castle and estate at Bailieborough and went to live on his estate at Hamilton's Bawn in Co. Armagh.

”The new owner of the Bailieborough estate was Major Charles Stewart of whom we know very little. His son, William Stewart, was High Sheriff of Co. Cavan, and later M.P. for the county. On his death, his son, another Charles Steward, a Dublin lawyer, succeeded him. He, in turn, became M.P. for Cavan. He had the reputation of being a good landlord. He was killed in a street accident in Dublin in 1795, and his estate passed into the hands of his nephew, Thomas Charles Stewart Corry of Rockcorry, Co. Monaghan. Mr. Corry was a minor when he inherited the Bailieborough estate. He never took much interest in the estate. In 1814 he sold out to Colonel William Young of Loughgall, Co. Armagh.”

Even better!
______________________________________________

A deed in the Bodleian:

Davys to Ormond
Written from: Dublin

Date: 19 October 1664

Shelfmark: MS. Carte 33, fol(s). 644

Document type: Holograph

Further particulars concerning the vacant see of Clonfert & Kilmacduagh (now void by the death of William Bayly, D.D., late Bishop thereof. ...
_________________________________

And another deed to get his death date even nearer:

Sir Paul Davys to Ormond
Written from: [Dublin]

Date: 10 October 1664

Shelfmark: MS. Carte 33, fol(s). 626

Document type: Holograph

It is written hither but now, that the Bishop of Clonfert and Kilmacduagh [who died, it would seem, on the day following. Cotton, by an oversight, says "August" instead of "Oct" (Fasti, iv, 168).] is at the point of death. ... The see is said to be worth £1,000, per annum, with a noble seat at Clonfert, near the Shannon and Portumna. ...
______________________________

From Fasti Ecclesianae Hiberna, vol 4, pp. 167-8:  (Bishops of Clonfert)

1644. William Bayly, D.D., a native of Scotland, educated at Glasgow (but D.D. of Oxford), having been driven out of his country by the Covenanters, fled into Ireland and afterwards joined King Charles at Oxford.  He was promoted to these sees by patent dated March 23rd; and was consecrated at Oxford, by Archbishop Ussher, assisted by the Bishops of Killals and Down, on May 2nd.  It appears from the Rolls of Chancery, 18-20 Car.I. that he had been designed for the see of Kilmore, upon Bishop Bedell’s death; for we find a revocation of several letters patent made to him, dated March 3rd, 1643-4.  [Todd’s MSS.]  The Bishop had little enjoyment of his see until the King’s Restoration.  He died at Clonfert, on August 11th 1864, and was buried in the cathedral.
_______________________________