So I assume when you start the Mapillary app, that you get the screen with a Sign up and a Login button. You press the Login button.
Do you then get a choice for how you want to log in? Facebook, Google, Regular Email, OpenStreetMap, Log in with ArcGIS. Which of these do you use?
Do you get a error message when you try to log in? The error message may be small and only on the screen for a few seconds. For example, when I press Google to log in (I didn't connect Mapillary to Google) I get a small message in the bottom of the screen "Could't connect to Google" that is only visible for 2 seconds.
And of course you need a functioning internet connection. Try to open the Mapillary webpage from your mobile phone to confirm that you don't happen to be outside of cell- or WiFi-cover.