The meaning is
Her nose, which was looking like a squashed tomato
At this, Winky howled even harder, her [nose (resembling a squashed tomato)] dribbling all down her front, though she made no effort to stem the flow.
"His/her something of a something" is a phrase commonly used to compare a face feature with some object. Examples from Google Books found using "his * of a nose/eye/ear":
His black dot of a nose rested between his paws.
If they knew that his one grave slit of an eye had seen gold in rice grains and furtive wealth in human lives, they were not prepared to abuse his memory, because Lung had stolen from them without vulgar ceremony
Mad Jack stroked his chin, began fidgeting with the ring in his gnarled lump of an ear.