There is nothing wrong with the solution offered in the video, the instructor simply took a few things for granted and assumed that everyone would understand (as is common in JEE preparation) . During collision, the short-lived contact forces generated between the two bodies are of a much higher magnitude (generally) than the regular forces such as gravity, buoyancy, etc. As such, in the presence of such forces, gravity does not play much of a role. Thus, during collision, gravity may well be neglected and the conservation of momentum is reestablished (for a more convincing proof, look at how the conservation of momentum was derived and try to use the impulse equation used in the derivation. You will find that the error is marginal, so small that it can well be neglected).
@Sahil's second doubt: The velocity of the plank is not specified at a single instant, the question merely says it is 10m/s. Thus, it means, by some contrivance the platform speed is kept constant. It is not for us to think what that contrivance is (a force equal in magnitude to its weight could be acting on it. The platform could be of a mass very large compared to the ball, so that the impulse contact force makes negligible impact on its momentum. There could be a hundred different ways to do it) but to assume the platform behaves kinematically (like a "wall", which remains as it is even with application of a net force). As for the ball, the velocity with which it hits the platform is given to us, so there should be no problem.
Alternatively, the velocity of the ball can be found by changing the reference frame to the platform, so it becomes a standard "wall". The ball approaches it at 15m/s (relative) and departs at the same speed (again relative). Move back to ground reference and v = 10 + 15 = 25m/s. After that, the rest is clear.
Yes you are right. The momentum cannot be conserved because of the action of gravity as an external force.
So it seems like the question is flawed, or was not framed properly.
(It also said the plank is moving up with a const. velocity and the ball also falls on it with a const velocity. How can we have a constant velocity when there is a total force g downwards? So, looks like there's some problem in the ques.)