I didn't understand why exactly the aliens wanted Earth's nations to cooperate in order to receive.
It seems the gift was the language which allowed one to see the future, but Abbott and Costello gave this to Louise.
Why would they do this if they intended all nations to work together in order to get it?
What prevented the scientists of other nations from getting it as well?
I will respond from my opinion:
The aliens will need humanity's help in 3000 years. They know they won't get that help if humanity destroys itself before that time. That's why they want to teach humanity to cooperate, "before" getting the gift, because after they get the gift, humans don't need more outside help, since they already will know the bad outcomes from not cooperating in general.
Louise didn't have the gift "before" it was given to her, because "before" she was awarded the gift, she still lived in a linear time. After she gets the gift, her time is not longer linear, it's circular. And in a circular time, there's no before and after, all times are the same and all times coexist.
Louise was willing to cooperate with other humans and with the aliens. They needed a cooperative person.
Perhaps what prevented the scientist from other countries to get the gift directly from the aliens too, was that they were not so willing to cooperate and still didn't really trust each other. But the gift was for all humanity in the end, not just for Louise. Anyone who reads her book, gets the gift too.
I hope this helps.
It seems the gift was the language which allowed one to see the future, but Abbott and Costello gave this to Louise.
Abbott and Costello were the aliens in the Montana ship. They gave it to Louise because she was the human they were interacting with. Aliens in the other ships would've been trying to give the same information to the humans they were interacting with, and may very well have succeeded too; we just don't see it on screen.
Why would they do this if they intended all nations to work together in order to get it?
As above, the aliens in all the ships are working towards the same goal with the humans they're talking to. Louise is just the one who got there first.
The aliens want all of humanity to work together because they need humanity to survive the next 3000 years without obliterating ourselves. But that doesn't mean that everyone needs to figure it out at the same time - just that the gift be transferred to humanity in general (which Louise ultimately does by publishing her book on their language.)
What prevented the scientists of other nations from getting it as well?
Nothing. For all we know, scientists in some or all of the other nations did figure out the language as well. The movie sticks with Louise and her team, so we don't really see what the others did or did not learn by themselves.
I certainly don't recall anything suggesting that people other than Louise failed to learn the language. It seems like the Chinese general probably didn't learn the language himself (based on his saying "he can't understand how [Louise's] mind works" in their meeting 18 months in the future) - but he's just one man. All that tells us is that he, personally, did not take the time to learn this new language. Other people around the world may very well have done so by then, for all we know.
Edit: Ian probably didn't learn the language either, otherwise he wouldn't have needed to be told what would happen with their daughter.
Just before the bomb went off, Abbott and Costello spewed a great many symbols. Ian eventually figured out that it was 1/12th of a message. They needed to cooperate with the other nations to obtain the complete message.
We the audience never learn what the complete message is. It seems likely it was important. Writing it was one of Abbott's final acts. Louise had learned enough of the language to start remembering pieces of her future, but that doesn't mean she had completely mastered it. The complete message may have been a guide to the rest of the language. Or, it may have been something else that humanity would need to know to be able to help the aliens 3,000 years in the future.
