How do phones charge their battery at seemingly any rate with no issues? 500mA or 1A or 1.2A depending on the charger?
'constant current' just means the charger will act like a current source and provide the cell with a specific amount of current, at whatever voltage the cell wants to sit at.  The current doesn't have to be 1A or 2A or whatever, the charger will use the largest current that it can get away with drawing from the source that the phone is connected to.  I would imagine the charger has a configurable current limit, and the phone's CPU will adjust this depending on how much power it can request from the source.