Oxford Dictionary of National Biography:
"Bateson, Thomas (d. 1630), composer and organist, first appears as the father of a son baptized in 1592; he may have been born in the Wirral, Cheshire, but the identity of both his parents and his wife is unknown."
- ODNB (copyright so not reproduced in full) says more about his career but very little about his family, it conlcudes:
"Bateson made his will on 2 March 1630 and died, in Dublin, a fortnight before his quarterly rent was due (which would have been, presumably, on 25 March); his widow was allowed a new lease on 30 April. He may have been buried in Christ Church Cathedral."

from Anne: A reference to the children of Thomas Bateson 1570, was in a 1928 edition of Groves Musical Dictionary.  The current edition does not give their names.
 
His children were Thomas 1603, Jane 1605, and Sarah 1607, no mention of a wife.  Richard Betson, it says, may also have been a son from an earlier relationship (presumably the one mentioned by ODNB).  These were taken from St Oswalds Register, Presumably this would be Chester and would pre-date his departure to Dublin.  To be certain, this would require some research.

IGI: Thomas Bateson Birth: 1570 Wirral Of, , Cheshire. Death: MAR 1630
- this seems to have been copied from the ODNB rather than an original record.
THOMAS BATESONNE Christening: JUL 1584 Mobberley, Cheshire (no father given)
JOHN BATESONNE Christening: OCT 1585 Mobberley, Cheshire
Father: THOMAS BATESONNE
(Mobberley is south of Manchester airport - not on the Wirral...)
If he was b.1570 he would only have been 14 in 1584....
THOMAS BATESONNE : spouse ALICE LEIGH Marriage: 09 NOV 1584 Mobberley, Cheshire

If the details of his children are correct then he would have married (his 2nd wife?) nearer 1600, but no record of that found in Cheshire - several in other counties and he could have moved from anywhere to take up the prestigious post at Chester Cathedral.

a possible daughter:
Jane Bateson Christening: MAY 1604 St Olave's, Chester, Cheshire
- St. Olave's records are available from 1603 with no sign of the others there or anywhere else in Cheshire.

By 1609 Thomas Bateson and his family were in Dublin but it is not known what happened to any of his children, including whether they remained there or returned to England. In 1649 Thomas Cromwell led his brutal reconquest of Ireland. To what extend this might have affected the Batesons if they were still in Dublin, or the parish records of them, needs further research.

IGI: Thomas Bateson Birth: 1680 Dublin, Ireland (no other details given)
from Henry: "My uncle told me that the Bateson's came from Ireland," - this is the only one in Ireland within 20 years. It is just possible that the composer Thomas was the gg-grandfather of Thomas b.1680, and could explain a possible birth in Dublin.

This one: THOMAS BAITSON Christening: 24 MAY 1674 Otley, Yorkshire - appears to have married Mary Appleton in Otley, 1697. No other likely candidates on IGI in Yorkshire to fit the one below:

from IGI and Mike:

Thomas Bateson
Birth: between 1670 and 1680
Death: September 1744; Buried 7 September 1744 Armley Churchyard, Leeds (see sketch right, click link for more details)
Armley Chapelry Records, Burial Register entry 7 September 1744.

Mary Hardcastle
Burial: 10 April 1753 Armley Churchyard, Leeds
Armley Chapelry Records, Burial Register entry 10 April 1753.

Marriage: 3 February 1702/3 Headingley Chapel, Leeds
Headingley Chapel Records, Marriage Register entry: 3 February 1702/3.
IGI: THOMAS BATESON Spouse: MARY HARDCASTLE
Marriage: 03 FEB 1703 Headingley, Leeds, Yorkshire
Death: 07 SEP 1744
photo: Undated. Print copied 3rd July 1908, titled ` Old Headingley Church as standing up to 1837 by Miss Fanny Thompson of Headingley or Miss Goodall of Halifax`. An old chapel had stood on this site, church built in 1837, then completely rebuilt in 1884/5.
- from Leeds Photographic Archives (Leeds City Council) click link for larger pictures, comments and to buy prints.

Child 1. John Bateson b. 1703
John Bateson
Birth: 1703; Baptism 11 November 1703 Armley Church, Leeds, John Bateson 'of Wortley'
Death: October 1762 ; Burial: 28 October 1762 Armley Churchyard, Leeds
Armley Chapelry Records, Baptism Register entry 11 November 1703.
Armley Chapelry Records, Burial Register entry 28 October 1762.
IGI: JOHN BATESON Birth: 17 OCT 1703 Christening: 11 NOV 1703 Saint Peter, Leeds, Yorkshire Father: THOMAS BATESON (some confusion whether he was baptised in Armley or Leeds....)

Combining all the info on the composer Thomas Bateson:

• Born about 1570 parents unkownIdentity of first wife unknown Son Richard b.1592, presumably on the Wirrall leading to the idea that Thomas was born there. 1599-1609 organist at Chester Cathedral, also employed in mending the organ.Children (from 2nd wife?): Thomas 1603, Jane 1605, Sarah 1607. 1604 published first set of madrigals. By March 24 1609, moved to Dublin where he was organist and vicar-choral for Christ Church Cathedral. 1612 (or 1615) became the first person gain BMus degree from Trinity College Dublin.1618 - second set of madrigals and Master of Choristers. 1622 became MA Made his will on March 2, 1630, DublinDied, probably, March 11, 1630, Dublin
• Possibly buried in Christ Church Cathedral

Wikipedia: Thomas Bateson, Batson or Betson (c.1570 - 1630) was an English writer of madrigals in the early 17th century.He is said to have been organist of Chester cathedral in 1599, and is believed to have been the first musical graduate of Trinity College, Dublin. He served as Vicar Choral and organist of Christ Church Cathedral, Dublin from 1609 until his death. He is known to have written church music, but only one of his anthems has survived, a seven-voice composition entitled "Holy, Lord God Almighty". His fame rests on madrigals, which give him an important place among Elizabethan composers. He published a set of madrigals in 1604 and a second set in 1618, and both collections have been reprinted in recent years. He died in 1630.
(text mostly copied from the 1911 Encyclopedia Britannica)

Some of Thomas Bateson's music for free download:
www.oldmusicproject.com/.../bateson.html

from: www.classical-composers.org/comp/bateson :Music
Thomas Bateson wrote church music but is chiefly remembered for his madrigals, of which he published two volumes — the First Set of English Madrigals (1604, 29 songs for 3 to 6 voices) and Second Set of English Madrigals (1618, 30 songs), both reprinted in “The English Madrigal School”, volumes 21–22, edited by E.H. Fellowes — which included some fine serious pieces that show an excellent taste in poetry. His Madrigal “When Oriana walked to take the ayre” was composed for inclusion in Thomas Morley “The Triumphes Of Oriana” (1603) but not included in that famous florilegium. The quality of his madrigals is generally regarded as somewhat inferior to those of Morley himself and Thomas Weelkes. Life
This English composer was born in Cheshire, a central English county on the Welsh border. After serving from 1590 as organist at Chester Cathedral (the county’s capital) Thomas Bateson went to Holy Trinity (now Christ Church) Cathedral in Dublin, Ireland, as vicar choral (singer) and organist in 1609, and became master of the choristers in 1618; in 1615 he became — alledgedly the first — Dublin B.Mus, and M.Art (Trinity College, Dublin) in 1622. This contribution is based on N. Slominsky’s Concise Edition of Baker’s Biographical Dictionary of Musicians (8th ed. NewYork, Schirmer Books, 1994) and www.hoasm.org/IVM/Bateson.html from www.answers.com/topic/bateson-thomas: (b? c 1570-75; d Dublin, March 1630). English composer. Possibly from the Wirral, Cheshire, he was organist of Chester Cathedral from 1599 and from 1609 of Christ Church Cathedral, Dublin, where he graduated BMus (1612) and MA (1622) at Trinity College. His two madrigal books (1604, 1618), while not specially original, include elaborately scored, serious pieces in a distinctive vein.The ODNB article is by David Brown who gives as his sources:

• New Grove E. H. Fellowes, The English madrigal composers, 2nd edn (1950) J. C. Bridge, ‘The organists of Chester Cathedral’, Journal of the Architectural, Archaeological, and Historic Society of the County and City of Chester and North Wales, new ser., 19 (1913), 63–124
• H. W. Shaw, The succession of organists of the Chapel Royal and the cathedrals of England and Wales from c.1538 (1991)

• Chester Cathedral website Wikipedia on Chester Cathedral - includes list of organists: Thomas Bateson 1599-1609 Christ Church Cathedral, Dublin, website Wikipedia on Christ Church Cathedral, Dublin - also has list of organists: Thomas Bateson 1608-1631Trinity College Dublin website - and Department of Music
• Wikipedian on Trinity College Dublin - link to famous alumni lists Thomas Bateson.