The lack of refresh was due to the fact that I did not have the zip code and street fields filled.
Is it possible for shipping methods to be updated after the city selection itself, regardless of whether the zip code and street are entered or not?
Przesyłki wyświetlają się jednak w zakładce InPost na WordPressie, ale ze statusem 404 i w Manager Paczek na stronie InPostu ich nie ma.