The above list is interesting to me in that it lists J.C. Onions, later one part of the merger that formed Allday & Onions, but no sign of any Alldays.
It had puzzled me that the other merger party, William Allday and Sons, variously claims establishment in 1720 or 1780, but doesn't start to appear in the press or directories until about 1850. The presence of Thomas Ingley at 56 ½ Constitution Hill gives a clue as to the Allday company origins.
William Allday resided at 32 ½ Constitution Hill, and I have his parents as being Peter Allday and Phoebe Inghley. Further investigation suggests that Phoebe Inghley and Thomas Ingley were a brother and sister from Halesowen. Thomas Ingley of Constitution Hill actually died in 1836, and his will leaves his bellows-making business to his son, Thomas jnr. Thomas jnr., however, is consistently listed in later directories as a beerseller and shows no sign of having taken up his father's business, while William Allday is the only bellows maker I see on Constitution Hill in the 1841 census. This all suggests that the Ingley operation passed not to Thomas jnr. but to Thomas snr's nephew, William Allday, instead.
Wright's Directory of 1849 confirms:
"Allday William, (late Thos. Ingley,) manufacturer of smiths' and small bellows, 32 ½ Constitution Hill"
... so it seems that the origin of that particular branch of the Alldays and Onions bellows makers lies with the Ingleys of Halesowen.