It was originally 2 semi-detached houses.
According to my copy of Almost an Island
(NoŽl E Smith), the school that became Weatherhead was located at in the building from 1890. It goes on to imply they were still at this site in 1908 when Cheshire County Council took over running of the school. However, the section about St. Columba's Church on the Egremont Churches page of PaulWirral's site
suggests they had vacated before 1904.
The name seems to come from the original conversion to flats after the school left the premises, rather than a name applied to the two homes.