Thanks, M. Here's the 'AE' character if you ever wanted to copy/paste it. Æ
The building could have been privately owned - I imagine it was - and the name was just used from one generation of glassmaker to the next. It happened with the Vesta Works of Shakespear (1801), then Walsh Walsh (1850), and British Heat Resisting Glass Co. (making Phoenix from 1935). However, as Walsh Walsh was still in business until 1951, I am wondering whether Phoenix leased a part of the WW factory to start up their production of heat-resisting tableware.