I don't think it's a plot hole. However, the movie kind of implied to me that the condition of the Pride Lands is based on the goodness of its leader. But, I knew this couldn't be true because the king can't control the weather.
Anyway, to answer your question, I don't think it's a plot hole. Remember it started raining even before Simba ascended on Pride Rock, which would have replenished the waterhole nearby so at least they could quench their thirsty mouths. So, at least, the pride had something to stay at Pride Rock for.
Then, I would say about two weeks to a month, they saw fresh new green grass growing, so that would bring the herds back. This gives them another reason to stay in the Pride Lands. And then, when enough herds return, they can officially stay in the Pride Lands.
Nature just has a weird way of healing itself. Although, I don't think the Pride Lands revived itself instantly as the movie implies, but since the rightful king had returned, and things in the kingdom were going in the right direction gave the pride some hope that they can stay in their homeland.
Made by Julie Skywalker. Thank you very much!