How does the lift coefficient of an A380 change when flaps are extended?
As mostly when questions are vague, the answer is: It depends.

If nothing else changes, the nominal lift coefficient stays the same. A coefficient is referenced to an area, and for aircraft this is the reference area which is derived from the wing area. Regardless of flap settings, the reference area stays fixed, and so does the lift coefficient because neither the mass nor the speed of the aircraft have changed when flaps have been set.

In reality, the aircraft will slow down when flaps are extended, and it can afford to do so because drag is increased and stall speed lowered. Now, with the lower flight speed, the lift coefficient will go up in proportion to the ratio of the speeds before and after squared:
$$c_{L_{flaps}} = c_{L_{clean}}\cdot\frac{v_{clean}^2}{v_{flaps}^2}$$
Again, this assumes the same reference area.

In a wind tunnel where the aircraft model is fixed to a sting and the angle of attack is controlled, setting flaps will increase the lift coefficient by the difference in zero-lift angles of the aircraft with and without flaps, multiplied by the lift curve slope of the aircraft.

$$c_{L_{flaps}} = c_{L_{clean}}+ (\alpha_{0_{clean}} - \alpha_{0_{flaps}})\cdot c_{L_{\alpha}}$$

Note that the lift curve slope increases when setting flaps increases the effective wing area. Again, all is referenced to the unchanged reference area.