In most cases angular cheilitis will go away on it's own without any treatment.
But you can help get rid of the angular cheilitis faster and get relief sooner by applying some Lip balm or thick emollient ointment frequently to the affected area.
Topical antiseptics and even applying some apple cider vinegar to the area where the angular cheilitis is and it will help get rid of it sooner and make your feel better.
When I have angular cheilitis flareups I apply apple cider vinegar to it to help it heal faster and it usually goes away within a few weeks.