A. The CONSTABLE family goes far back at Wednesbury, but John BEDO is the first BEDDOW. Looking at the John BEDO-Mary CONSTABLE family, one finds some interesting early info in the Bishop's Transcripts (BTs) for Wednesbury: 
John BEDO mar. Mary CONSTABLE 29 May, 1732

Bedo, Jonathon s/of John & Mary; bapt. 8 Aug., 1732
  Bedo, Jonathon s/of John & Mary; bur. 9 Aug., 1732
? c. 1733
? c. 1735
? c. 1737
? c. 1739
Beado, Benjamin s/of John & Mary; bapt. 26 Dec. 1741
Beaddow, Danil s/of John & Mary; bapt. ? ???. 1743
  Beddow, Daniel s/of John & Mary; bur. 22 Oct. 1754
Beddow, James s/of John & Mary; bapt. 23 Jan. 1745/6
? c. 1747/8
? c. 1749/50
Beddow, Isaac s/of John & Mary; bapt. 8 Feb. 1751/2
Other Beddow burials at Wednesbury: 
Beddow, John    31 Dec. 1754
Beddow, Joseph  29 May  1765
Beddow, Mary            1784
Marriages with interesting permutations on witnesses at Wednesbury: 
10 Feb. 1750/1  Thos Malpas and Anne Beddow
23 Nov. 1757    John Burkett and Anne Wilkes; wit.: Thomas Malpas
 9 Feb. 1757/8  William Beddow and Hannah Yates; wit.: John Burkitt
20 Jan. 1761/2  John Beddall and Sarah Peers; wit.: Thomas Malpas
A search of the International Genealogical Index (IGI) at http://www.familysearch.org/ reveals a high concentration of BEDDOWs to the west in Shropshire. One group of baptisms at Broseley, Shropshire for a John and Mary Beddow seems very interesting: 
17 Feb. 1733/4   Ann
30 Jan. 1735/6   William
26 Dec. 1737     Joseph
26 Dec. 1739     John
fitting well with the gap at Wednesbury and accounting for those BEDDOWs recorded as parents of the next generation. The IGI works well as a finding aid, but in no sense is a complete extraction of all extant records. However, since there is no other conflicting family in the IGI and since these two mesh so well, a good line of research would investigate this as one family: 
Jonathon  bapt. Wednesbury  8 Aug. 1732;   bur. Wednesbury  9 Aug. 1732
Ann       bapt. Broseley   17 Feb. 1733/4; mar. Wednesbury 10 Feb. 1750/1
William   bapt. Broseley   30 Jan. 1735/6; mar. Wednesbury  9 Feb. 1757/8
Joseph    bapt. Broseley   26 Dec. 1737;  ?bur. Wednesbury 29 May  1765
John      bapt. Broseley   26 Dec. 1739;   mar. Wednesbury 10 Jan. 1761/2
Benjamin  bapt. Wednesbury 26 Dec. 1741;
Daniel    bapt. Wednesbury         1743;   bur. Wednesbury 22 Oct. 1754
James     bapt. Wednesbury 22 Jan. 1745/6; 
?                               c. 1747/8
?                               c. 1749/50
Isaac     bapt. Wednesbury  8 Feb. 1751/2
B. To be sure, BEDDALL and its variants are to be found in the Black Country; but the name is very rare at Wednesbury before 1813. The 18th century Bishop's Transcripts include the marriage of John Beddall to Sarah PEERS, a 1793 baptism for a child of John & Charlotte BIDALL and a burial of Luke BEDDALL (under 1), son of Thomas and Mary on 7 Nov. 1771. I would like to examine the PRs for that burial to compare with the baptism on 23 Oct. 1771 for Luke BEDDOW, son of William and Hannah. There are several 17th century entries of baptisms and burials for BIDDLEs (BYDDEL, BYDDELL, BYDDLE).
C. I am certainly relying some on the coincidence of the marriage of John Beddall to Sarah PEERS with the beginning of the John and Sara BEDDOW family as evinced in the baptismal record at Wednesbury, along with the lack of an alternative BEDDOW marriage as well as an alternative BEDDALL family. Further, there was coincidentally a marriage between a John BEDDOW and an Ann GENDERS on 13 March 1785, 8 months after the burial of "BEDDOWS, Sarah wife of John" at Wednesbury. This marriage was witnessed by Thos BLAKEMORE, the husband of one month to Lucy BEDDOW the daughter of John and Sarah.
D. The microfiche index for St Bartholomew, Wednesbury shows but one 18th century Malpas(s) family, that of Thomas Malpas and Anne Beddow, married 10 February, 1750/1. Their children: 
John       bapt. 1752
Elizabeth  bapt. 1754  died 1760 of smallpox
Anne       bapt. 1756  died 1760 of smallpox
Hannah     bapt. 1758  died 1765 (sic) of smallpox, but may have been 1760?
Thomas     bapt. 1760  died 1760 of smallpox
William    bapt. 1761  died 1765
Hannah     bapt. 1764    (Mary illeg. d. of Hannah  1784)
Thomas     bapt. 1767  died 1768
Samuel     bapt. 1769
Thomas     bapt. 1773
Mary       bapt. 1774
I believe Anne Beddow was the older sister of my John Beddow. Thomas Malpas witnessed the marriage of John Beddall (sic) and Sarah Peers at Wednesbury 20 January, 1761.
E. The Bishop's Transcripts for St Bartholomew, Wednesbury show another Beddow family, that of William Beddow and Hannah Yates, married 9 February, 1757/8. Their children: 
Anne       bapt.  8 Nov. 1758   buried 23 Nov. 1760
Sarah      bapt.  9 Apr. 1760
John       bapt.  5 Nov. 1761   buried 21 Jul. 1765
Hannah was buried at Wednesbury 16 October 1762 and it appears that William married Sarah Smith 20 November 1764.