Im playing as a Nord and i have noticed this too, i cant see any sort of timer either and it seems i can stay underwater for as long as i want.
That's what I thought at first but it turns out there must be a timer! After about 3-4 minutes underwater suddenly your health bar comes up at the bottom of the screen and you start taking damage (like in previous games). The problem is that it isn't visible when the timer is going down.
Also have you experienced the breathing sounds underwater just as I have?