After researching the origin of the name of each city, it seems the pattern is:
[Word that describes the theme of the town] + [(sometimes European based) Suffix that describes the physical nature]
References and sources is linked in this google doc.
Postwick
Post (Poss) + wick (wic)
Postwick could be based on the real life Postwick, who's name is derived from "Poss wic", meaning dairy farm. This could be interpreted as "Dairy" alluding to it being the hometown of the player, with "farm" being the physical nature of it being, well, a farming town. Alternatively, "post" could refer to starting post, which while a stretch isn't impossible with Sword and Shield's sports theme.
Wedgehurst
Wedge + hurst
Could possibly refer to Wedgehurst/the Pokemon Lab being in between the player’s hometown “Postwick”, and the players adventure to the Pokemon league/dynamax. "Hurst" means hillock.
Motostoke
Moto (Motor) + stoke
Admittedly the formula is a bit of a stretch with Motostoke. It is made up of 2 words, "Motor" and "Stoke". Both these words refer to the steampunk theme of the city, as well as the Fire type gym leader.
Turffield
Turf + Field
According to google, the definition of "turf" is "grass and the surface layer of earth held together by its roots." which is relevant in the grass type/farm theme it has going on. "Field" is a bit self-explanatory.
Hulbury
Hul (Hull) + bury
"Hull" refers to the hull of a ship, reflecting Hulburry's water theme. "Bury" is a suffix for a British town suffix for enclosure.
Hammerlocke
Hammer (or Hammerlock) + Locke
"Hammerlock" could refer to Hammerlock being in the center of Galar and home to one of the most powerful trainers aka Raihan? Or simply "Hammer" refers to his attitude as it's informal definition is "attack or criticize forcefully and relentlessly." "Locke" is a toponym for lockable enclosure, which likely refers to the vault which is locked to the public.
Stow-on-Side
Stow + on side
"Stow" likely references the hidden chamber stowed behind the mural. It may also refer to the nature of its gym leaders, Bea (who's emotional side is often stowed away so as not to show weakness) and Allister (who thanks to his anxiety, literally stows away his face from the public, as well as his sociability). "On-side" could come from the fact that Stow-on-Side is literally at the side of the Galar map. Might also be another sports reference.
Edit: According to SadisticMystic's comment, "On-side" is likely to actually refer to the city being along a river named Side, if we go by British naming conventions. So this further supports that the second word in the formula is based on physical attributes of the town.
Ballonlea
Ballon + lea
“Ballon” is a variation of the word Balloon, and also (in dancing) the ability to appear effortlessly suspended while performing movements during a jump. It could be representing the fairy type gym and the mystical nature of the town. “-lea”, being a British town suffix for woodland clearing, refers to the nature of the town being located within a forest.
Circhester
Cir (Circus) + chester
"Circus" are opened spaced venues in Ancient Rome. They were a venue for sports, which may be reflected in the gym challenge. Circuses are also venues for public events and theatres, which can be reflected in that Circhester is among the bigger towns in Galar and features locations like the Hero’s Bath, Hotel Iona, a hair salon, and a boutique. "Chester" is a British town suffix for fortification of Roman origin, which makes sense because Circhester is the home to the Hero’s Bath, which is probably a reference to its name’s Roman origin (rome region when?), which was often linked with heroes.
May also be a reference to Cirencester.
Spikemuth
Spike + muth (mouth)
"Spike", could be a reference to the town's "edginess", being dark type themed. And the punk rock theme. "Muth" could be a respelling of mouth, which refers to the cave like structure of Spikemuth, being a single corridor to the gym challenge.
Wyndon
Wynd + on (London)
"Wynd" is a word often used for narrow streets in the UK. Could refer to Wyndon's urban nature? The "London" part is obvious, its based on London. This may be an inversion of the formula, where the physical attribute goes before the theme.
Freezington
Freezing + ton
"Freezing" is kinda self explanatory. "-ton" is a British town suffix.
