Bridget's father is said by some to be the William BATEMAN whose will was proved 10 Oct 1658 in Fairfield, CT, but that will does not mention a daughter, Bridget.  Also, while we don't know where William BATEMAN, himself, was born, we do know that one of his children was born in Toppesfield, Essex, so it's unlikely he's the father of any children born in Hawkedon, Suffolk.

How do we know she was born in Hawkedon?  If we had a birth record for her, we should know her surname and the identity of at least her father. 
(The distance from Lavenham to Hawkedon, as the crow flies, is about 7 miles, and about 10 miles by road.)

1a.  Clarence Almon Torry.  1985.  New England Marriages Prior to 1700.  New England Historic Genealogical Society, Boston, MA (reprinted 1997, Genealogical Publ. Co., Baltimore, MD):

The "b" indicates the birth date of a child as an indication that the marriage took place before this time.  The locations are known residences of the husband, ending with the husband's place of death.  An entry in Torrey does not imply a marriage in New England; it only demonstrates the presence of a married couple there before 1700.  Torrey is a compilation of published sources, not primary records, so he should be considered a secondary, if not tertiary, source.

p. 109

BROWN, Thomas (-1688) & Bridget __?__; b 1642; Concord/Cambridge... BROWN, Thomas (-1690, 1691) & Martha (EATON) OLDHAM (1630-), w Richard; 7 Oct 1656; Cambridge

1b.  Clarence Almon Torry.  2015.  New England Marriages Prior to 1700.  New England Historic Genealogical Society, Boston, MA (repuslished; online at AmericanAncestors.org):

p. 256

BROWN, Thomas (-1688) & Bridget _____; by 1642; Concord/Cambridge {Brown (#3) 6; Sv. 1:276; Brown (1931(ms) 7; Fallass 106; Davis-Everett 44; Chandler (1949) 27; Morton (1947) 43}... BROWN, Thomas (-1690, 1691) & Martha (EATON) OLDHAM (1630-), w Richard; 7 Oct 1656; Cambridge {Cambridge 502; Watertown 202; Sv. 1:276; Brown (1931)(ms) 9; Eaton 275; Townsend (#6) 31; Reg. 76:73; LBDF&P #310; Morton (1947) 42-3}

p. 1110

OLDHAM, Richard (-1655) & Martha [EATON] (1636-), m/2 Thomas BROWN 1656; by 1649; Cambridge (Brigham Anc. says Martha Sherman) {Reg. 76:73: Brigham Anc. 41, 47; Brighton Hist. 198; Eaton Fam. Assn. 19; Watertown 202}

Even in this later edition, two Thomas BROWNs continue to be recognized by Torrey.

2a.  England Births and Christenings, 1538-1975 (online at FamilySearch.org):

Name	Thomas Browne
Christening Date	10 Jan 1605
Christening Place	Lavenham, Suffolk, England
Father's Name	Edmond Browne
GS Film #	402302

2b.  Massachusetts Vital Records to 1850: Concord Births (online at AmericanAncestors.org):

Vol. VI, p. 1	A register of the berths & burialls in Concord from the yeare 1639 untill the first month 1644 according to [?] account... Boaz the sonne of Thomas Browne was borne the 14°. (12°.) 1641.
Vol. VI, p. 5	Mary the daughter of Tho: and Bridget Browne, the (26) 1: mo: 1646 Eliezer the sonne of Tho: & Bridget Browne, the. (5) mo 1649:

These are the earliest births recorded in the history of Concord, and Boaz is listed 9th.
These are Old Style dates where the first day of the year is March 25th, so...
For Boaz, the 12th month is not December, it's February, and the year is now 1642, not 1641.
For Mary, the 1st month is not January, it's March, so her birthdate is now 26 Mar 1646.
For Eliezer, the 5th month is not May, it's July, so his birthdate is now July 1649.

3.  Sir John Bernard Burke.  1996.  "Burke's American Families with British Ancestry." Pages 2529-3022 from Burke's Genealogical and Heraldic History of the Landed Gentry (1939, Burke's Peerage, London).  Genealogical Publ. Co., Baltimore, MD.  On p. 2584:

BROWN formerly OF LAVENHAM... THOMAS BROWNE, of Lavenham, Suffolk, and afterwards of Sudbury, Massachusetts, whither he went with his brother, Edmund, bapt. at the Church of St. Peter and Paul, Lavenham, 10 Jan 1605, and d. 3 Nov. 1688.  His son,  ELEAZER BROWN...

I've found the baptism, but where did Burke get the death date?

4.  James Savage.  1860-62.  A Genealogical Dictionary of the First Settlers of New England Showing Three Generations of Those Who Came Before May, 1692.  4 vols.  Little, Brown, and Co., Boston, MA (online at Ancestry.com and GoogleBooks.com; color and boldface added; please note that this work is undocumented and therefore a secondary source):

p. 276

THOMAS, Concord 1638, perhaps br. of Rev. Edmund, being among orig. proprs. of Sudbury 1637, freem. 14 Mar. 1639, by w. Bridget had Boaz, b. 14 Feb. 1642; perhaps Jabez, 1644; Mary, 26 Mar. 1646; Eleazer, July 1649; and Thomas, 1651.  His w. d. 5 Jan. 1681, and he rem. prob. to Cambridge, d. 3 Nov. 1688. THOMAS, Cambridge, m. 7 Oct 1656, Martha, wid. of Richard Oldham, had Mary, b. 28 Apr. 1658, d. young; Mehitable, 13 May 1661; Mary, again, 1 Nov. 1663, prob. d. young; Ebenezer, 15 Jun 1665; Ichabod, 5 Sept. 1666; and Martha, 19 Oct. 1668; and d. Dec. 1690.  His will of 23 Nov. preced. ment. the w. and four ch.

I've found birth records for children Boaz, Mary, and Eleazer, but not Jabez or Thomas.

5.  Charles Henry Pope.  1900.  The Pioneers of Massachusetts. C.H. Pope, Boston, MA (republ. 1965/98 by Genealogy Publ. Co., Baltimore, MD; Broderbund CD-504; online at Ancestry.com; Kindle e-copy sucks; paperback copy from NEHGS):

p. 75

Thomas, Sudbury, propr. 1640.  The Gen. Court 7 (8) 1640, granted him 200 acres of land for the 25 li. adventure of Mrs. Anne Harvyes.  He seems to have rem. to Concord, where by wife Bridget he had ch. Boaz b. 14 (12) 1641, Mary b. 26 (1) 1646, Eliezer b. 5 (3) 1649.  He is clearly the man who was adm. chh. Cambridge May 18, 1666; had m. before 1658 Martha, widow of Thomas Oldham; ch. Mehitabel bapt. June 2, 1661, Mary bapt. March 8, 1663, Ebenezer bapt. July 23, 1665, Ichabod bapt. Sept. 9, 1666. [Mi.] Will dated Nov. 23, 1690, prob. in Jan. folg., beq. to wife Martha, daus. Mehitabel and Martha, and sons Ebenezer and Ichabod. The inv. shows house and land in Camb. and 300 acres at Worcester.

It appears Pope may be the source of the error equating the two Thomas BROWNs.  It's not insignificant that he omits death dates for both Bridget and her husband because those would have pointed up his error in equating the two men.  Note, also, that none of Bridget's children are mentioned in the will.

6.  Brian Freeman.  Our Family History and Ancestry (online at family.dumhop.com):  the pedigree for Thomas is undocumented, and I'm told by the site owner that the information was obtained from Sonia Abate.

7.  WorldConnect / Ancestry World Trees (online at RootsWeb.com/Ancestry.com).