The primary clue to the identification of this chap is from the marriage record of William Bartholomew and Mary Ann Seal in which the groom records his father as "William Bartholomew, labourer".  The other clue is that William B., father of Sarah B., always recorded his place of birth as Scotland; so that is where his parents must have come from.

On the Scottish Parochial records database there are two couples who had Williams for children about 1818 and of whom the father was named William:
(1) William Bartholomew and Mary Moore of Stanton, East Lothian who had a William baptised on 2 Nov 1817.
(2) William Bartholomew and Mary Brodie of Lochwinnock, Renfrew who had a William baptised on 14 Feb 1819.
Might this William be one of these?

Further investigation has shown that Will Bart and his wife Mary had a further child or two in Kentford, Cambs by 1821  So a scan of the Scots Origins database has provided only the one child of Wm and Mary Moore but three children of another Wm Bart and Mary Brodie from 1816 to 1823.  This leads to the conclusion that Mary Brodie could not have been the one to go down to Cambridgeshire and thus that (if the Scots origins records are complete) Mary Moore is my g-g-g-grandmother.

The birth / registration place for Wm, my g-g-grandfather is Stenton. (Where?)

Also on the Scots Origins database I found a marriage record for Wm Bart and Mary Moore:
  23 March 1816 at Dirleton,
     Scots Origins serial number 6524882, LDS film No FR723.

I could not, from Scots Origins, satisfactorily locate the births of either Wm Bart or Mary Moore.
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William was a Scotsman.  This is clear from the following censuses:

1841 where he said he was not born in Suffolk but was born in the British Isles but outside England.
In 1851 where he is not to be found but where his son, also William, said he was born in Scotland.
In 1861 he was dead but his son william still reported he had been born in Scotland.
In 1871 again his son William reported he was born in Scotland.
In 1881 finally, before his death, his son still reported he had been born in Scotland
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The other clue is that there were two separate families of Bartholomews at Kentford, Suffolk (a mere three miles away from Chippenham and ditto from Ashley).  Might they have been both descended from this original Scottish couple?

If William, husband of Mary Ann Seal, was born in Scotland then it is likely that his parents were married in Scotland.
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Now for this William’s birth:

On FamilySearch there are 4 Wm Bartholomews born in Scotland between 1760 and 1780, ten years either side of William’s birth year given in the 1841 census.  The identity of his parents is not known as they were not listed on their marriage register.

Jan 2016, TFPL: I have revisited the Scotland’s People website and now find there are three William Bartholomew births between 1760 and 1780:

   Parent Names              City/  GROS
No Date   Surname  Forename   Frame No.  Sex Parish  County Data

1  16/03/1761  BARTHOLOMEW  WILLIAM  JAMES BARTHOLOMEW    GREENOCK        564/01
                                     MARGARET FORREST   M  NEW OR  /RENFREW 0010
                                       FR150               MIDDLE           0223
2  03/02/1765  BARTHOLOMEW  WILLIAM  JAMES                                 486/00
                                     BARTHOLOMEW/  M  MUIRAVONSIDE /STIRLING 0010
                                     JANET GOW FR106                         0195

3  9/02/1773  BARTHOLOMEW  WILLIAM JOHN
                                      BARTHOLOMEW/                         491/00
                                      ELIZABETH    M  STRATHBLANE/  STIRLING 0010
                                      SMITH/ FR657                           0245
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No sign of him in the 1851 census in England or Scotland.  I have gone through all the enumerator’s sheets for Kentford.  Perhaps he was in a workhouse?
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Jeremy Bartholomew, a co-descendant of this William, has suggested that this is his birth:

                           S
Parent Names    e  Parish  City/   GROS
  Date Surname Forename Frame No.     x           County  Data

  PETER BARTLEMAN                         705/00
  /AGNES LIGO               /EAST     0020
01/09/1771  BARTLEMAN  WILLIAM  FR392 (FR392)   M   DIRLETON   LOTHIAN   0249

Apparently ‘Bartleman’ is a known alternative to ‘Bartholomew’.

There is no other baptism for any Bart* in the county of East Lothian, where Dirleton resides.

There are two baptisms in the whole of Scotland for this wildcard but the surnames are Bartlett and Barton, so they can be discounted.

There might be a unrecorded birth of other people but this has the added relevance that Dirleton is where he married at the age of 25.  Brillian!
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His death index:

Surname   Given Name   District   Volume   Page
Deaths Mar 1855
Bartholomew William Newmarket 3b 431
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His burial register entry:

Year:      1855
Nams:      William Bartholomew
Abode:   Kentford
When buried: March 2 1855
Age      85 yrs
Officiant:   S Charlton, Curate

From this he was born in 1770 plus or minus a year.
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The record is:

"William Bartholemew and Mary Moore both in this Parish gave in their names for proclamation March 23rd 1816 and were regualry proclaimed. Witnesses Joseph Alland and James Henderson."

Marriage would have been some days after this.

Dirleton is a village and parish in East Lothian to the east of Edinburgh.