My guess is that the Galwegians moved to Glasgow in large numbers at some point in history, bringing their dialect with them, and eventually Glasgow Galwegians comprised the majority of Glasgow residents. Then the name became associated only with the city when Galloway no longer existed as a political region. Perhaps there is a Scotsman out there who could confirm or refute that?
The inhabitants of Bristol are, I belive, 'Bristolians', whereas 'Bristolites' seems more natural.
According to a meander through the OED, Glaswegian is modelled on Galwegian (a resident of Galloway), which is modelled on Norwegian (a native of Norway) which derives from any one of a large number of variants such as Norwegia, Norwege .... old names for Norway.
What it doesn't explain is why an ending that is appropriate for places ending in -way came to be applied to Glasgow.