There's not a lot of info around on the brickworks, but I've managed to piece together a bit.
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Description: Wirral Patent Brick WorksDescription: 1909 OS MapDescription: Liverpool Mercury 3/11/1897
Gore's/Kelly's Directories reveal that at one point there were two brickworks on Borough Road at about the same time, possibly connected with the construction work going on in Devonshire Park where most of the houses were built at about this period. The main one, as you rightly say, was on the site later occupied by the Technical College, the other was further along on the same side, just before the bottom of Mount Road. This second one was established in mid-1882 by Llewellin Godsell, a coal merchant who owned coalyards in Cleveland Street and Exmouth Street; the directories indicate that the Cleveland Street yard had at one time also been used as a brickworks.
L Godsell & Son's Patent Brick Works is listed in Gore's/Kelly's until 1890, but not thereafter.
The one on the Tech site was known as the Wirral Patent Brick Works, and was built in early 1883 by Messrs Crowe & Williams (The plans for the works were approved on 30th November 1882). William Crowe and Thomas Williams were originally timber merchants, and owned two large sawmills in Birkenhead, the Birkenhead Sawmills in Bridge Street and the Victoria Sawmills in Price Street. Their names appear in the trade directories until 1888, after which the brickworks appears once in the name of William Dawson in 1890 before disappearing from the record. Local newspapers confirm, in fact, that the firm of Crowe & Williams was wound up in October 1887.
The association of your great-grandad's firm with the brickworks is puzzling, as the trade directories don't record any brickmaking taking place on Borough Road after 1890. Newspaper adverts (see below) indicate Jones & Bancroft, builders, were operating out of their original premises at 134 Oxton Road until at least 1897. By 1899, however, they had moved to 3 Kingsland Road, which of course backs onto the brickworks site. Perhaps they were merely using part of the old brickworks premises as a contractor's yard rather than actually making bricks there?
As diggingdeeper says the Wirral Patent Brick works isn't marked on any of the published OS maps, probably because the existence of the works falls between the 1875 Edition and the First Revision in 1899-1900. However, I have managed to locate a copy of a large-scale 1:500 OS sheet on which some kind person in the Borough Engineer's Department in the late 1800s has drawn on the location of the brickworks and its associated kiln. It's not the clearest copy in the world, but it's probably the only one that exists! As the scale is rather large and the section I've been able to copy shows little else apart from Valley Lodge (now McMullen's Funeral Home) at left and the bottom of Kingsland Road at right, I've also scaled it down and superimposed the buildings onto a copy of the 1909 1:2500 sheet to give a better idea of the location.