Many thanks for your research LizzieL, it's very kind of you to take the trouble. As you say any daughters would have been too young to go with the coachman. Wills were going to be one of my next steps..
My tree is solid back to George Horrocks ( B 1814) whose father was ( I think) John Horrocks. Prior to George, things get hazy since we're reliant on Parish Registers and other sources. I am guessing George's father would be born around 1790 or so. 1783 is a little early albeit plausible, although they tended to have children very early.
My difficulty is how to find a link between any Horrocks in Lancashire and those in the Birmingham area. Simple birth or marriage dates are not adequate to prove a person born in Lancs is the same one that married or died in Birmingham. If the daughter was cut off, the coachman presumably being sacked, (unless he was someone elses) they would need to find work of some sort. Migration out of the area would have been one option, although the 'pull' of Birmingham might not have been adequate at the turn of the 18/19th C.
One of those things it's difficult to get a grip on.
[EDIT]. Burscough's book on the Horrocks does include a series of pedigrees and I wonder whether it might be more fruitful to explore these to see if there are any plausible links.