@Emil I managed to solve the issue.
But first let me add some extra info:
The button was blinking red twice from not having been in touch with the Hub since I replaced the battery 3 days ago and the button was now listed as disconnected in the Hub view in the app.
I then removed the battery, waited a bit and then inserted it again.
The button was still listed as disconnected. I then pushed the button and it gave a single red blink and now the button moved from the disconnected section to the connected section in the app.
However, each time I pressed the button nothing would register.
Normally the app displays that the button vibrates when it has been pressed - but this time nothing happened beside the initial registering of it being connected.
Before "nuking" the button I tried removing the power cord from the hub (waited 10 secs) and inserted it again - but this didn't change anything.
I then took a backup of the webhook-calls for the button and removed it entirely from the hub.
Then I selected "Add a Flic" and followed the procedure for adding a new button.
The button was found and now it was detecting the button was pressed by showing the vibration animation.
Then I added the webhook calls back into the button and now it works just as before.
So I guess the solution was to remove the connection/button in the hub for it to properly reinitialize after having changed the battery.
Of course this isn't meant to happen - but I guess it does in some instances.
In order to mitigate how big a problem this is, I have a development request:
Could you please add a backup function in the app, which stores the functions of the buttons and which you can give a name (default name would of course be the name of the button that you backed up).
Then you can restore a button (or copy to a new button) from such a backup. AND it is of course important it only stores the function/button press values and not the connection-data, since this is what we are trying to reset.