Well let's take mundo's w or ulti. Would the damage of the w increase when you have it activated and you are leveling it up at this moment ? and if you activated his ult and you apparently hit lvl 11 and skill it up would you heal more for the rest of the duration ?
The answer to your first question is yes, since it is a toggled spell, the damage from Dr. Mundo's Burning Agony
W does in fact scale up immediately. Dr. Mundo's ultimate (Sadisim) on the other hand, is a triggered skill. As such, when
R is pressed, it applies a buff then is done. This means that the healing will not increase until the next time the skill is triggered.