Hi Philip, I've managed to do a bit of research concerning Conway cottage, but info on the History
of the house is hard to find. The exact date the house was built isn't known, but it doesn't appear on any of the maps of Birkenhead from the mid-1840s. It is, however, clearly marked on Mills & Fletcher's map of 1858 (see pic below) next to the K in 'BIRKENHEAD'. A block of property on this site seems to be marked on an earlier map from 1850 showing J M Rendel's plans for the proposed Birkenhead Docks, which would suggest that it was built some time in the late 1840s, possibly about the time that Birkenhead Park opened in 1847. A number of substantial detached 'villas' were built along Conway Street at this time, but most of them only survived 35-40 years before being demolished to make way for terraces of workers' housing.
The earliest occupant I've been able to trace is Alfred Dudley esq, who is listed in the 1851 Census, 1857 Post Office Directory & 1860 Gore's Liverpool Directory. He was originally from Kingswinford, Staffordshire, and is variously described as a commission broker or merchant, working for G Wheatley's India & China Express Company, whatever that may have been! He seems to have moved shortly afterwards, for by the 1861 Census he was living in the Manor House, Moorside, Neston.
I haven't had time to trace every subsequent occupant of the property, but in 1881 the occupant was James Harkness, a widowed builder from Scotland, and in 1901 it was John Ambrose Thomas, a team owner from Eastham. If you want to have a go, there are various sources available: Trade Directories (available at Birkenhead Reference Library or on microfilm at Liverpool Record Office, but sometimes unreliable), Census Returns (available online, but difficult to search if you haven't got a name to look for), Voters' Lists/Electoral Rolls (at Birkenhead Library & Wirral Archives), and Rate Books/Valuation Lists (Wirral Archives, but survival is patchy).
I have managed to track down some stuff relating to your grandfather's time at the house. He and his brother (?brothers) appear to have arrived there in early 1922, when they put in an application to build a temporary masons' workshop at the side of the house. This presumably soon became a permanent building, for the business of Alldis Brothers was well-established by 1928. In that year they applied to build a showroom on the boundary of the property fronting onto Conway Street. This was originally going to be built directly next to the pavement, but apparently the dentist next door at no. 166 objected, so the plan was revised and it was built six feet further back. I've included a photo of the showroom in the early 1960s; it's not great quality, but it brings back memories of walking down that end of Conway Street when I was a kid. God knows how many times I must have passed the place, but, although I remember the gate posts, I don't recall Conway Cottage itself - basically because you couldn't see much of it unless you were on the top deck of a passing bus!!
Hope this stuff helps. If you want copies of the surviving building plans, they're available at Wirral Archives, off Shore Road near Woodside Ferry.(If you can't get there in person you can email any requests to firstname.lastname@example.org)