It won't affect your piercing. Chlorine itself is bad for the skin but in a pool it's a form of cleansing, that's why it is important to have enough but not too high a level of it in pools to prevent the spread of germs as it disinfects the water in which we swim. Just make sure after you've had your swimming lessons you go home and boil water, put sea salt in it and let it cool a tad then soak the water on your nose with a cotton pad and the sea salt and it will be fine.
