There is a pretty well-hidden proxy setting in the WLAN settings that was set to Enabled behind the user’s back. Long-press your Access point, choose the lower one (Tiedot in Finnish, could be Information) then choose Edit from the top pulley menu and set proxy to Disabled.
I had this on 4.5, 4.6 and 5.0. On the Sailfish side checking for Sailfish updates also complained about not having an Internet connection.