See Distribution of Balshaws, Belshaws and Bolshaws, 1601 to 1850.
In the Cheshire IGI there are more Bolshaws than Balshaws, and very few Belshaws. However, there are several cases of different spellings being used for the same families. Does this suggests that the name was pronounced as Bolshaw in this area.
In Lancashire there are more Balshaws (especially in the north of the county) than Belshaws (who are predominantly in Wigan, Manchester and Merseyside). There are a few Bolshaws, (mainly around Manchester and >1700) but it seems likely that the original name was Balshaw and that Belshaw and Bolshaw are variants of it. Most of the Belshaws seem to have come from a relatively small number in the area around Wigan.
In the Yorkshire IGI there are 9 entries for Balshaw (in Hecke, Snaith, Gt. Milton, Thornaby, Wakefield and three in Slaidburn). There are nearly two pages of Belshaws, mainly in the North and East Ridings. There are four entries for Bolshaw, (in three in Bradfield and one in Sheffield).
In the N Ireland phone book (1995) there are no Balshaws or Bolshaws. There are 62 Belshaws, including 12 in Belfast and 22 in Lisburn). There is a Belshaw's Quarry nature reserve just west of Lisburn.
Edward Pearson Balshaw apparently claimed not to be related to the 'Cheshire Balshaws'.
Run by Fraser Stewart, who says:
I recently received an old diary from my father that contained a lot of BELSHAW`s. Also with this he gave me a folder that contained a bit of information on my family tree. My father had paid for a geneologist to do some research for him in 1977. There were no BELSHAW`s mentioned in the documents but mostly it was my direct line of ancestors "STEWART".
Thanks to the internet I have managed to find out a little bit of information about the BELSHAW`s mentioned in my old family diary .They tie into my family tree in 1915 when AGNUS BELSHAW married my great grandfather JOHN HENRY STEWART. I have a family tree website. On this website I have information on around 30 BELSHAW`s in my family tree nearly all situated in the Hetton-le-Hole, Houghton-le-Spring area of county Durham, dating from 1951 back to 1799 in Ireland.
I am very keen to learn anything I can about my ancestors, no matter how distant.