How did Bucky originally lose his arm?

A Soldier and a Wolf. Found alive, though missing his left arm from the fall from a HYDRA train, Bucky Barnes was placed in the hands of Arnim Zola.

When did Bucky get his arm cut off?

I think in the mcu, he basically lost it in the fall from the train and it was replaced when Zola experiment on him. In one of the winter soldier flashbacks, I think we see Hydra doctors removing it via surgical saws and stuff.

Who gave Bucky first arm?

In one of the special’s closing scenes, Nebula gives Rocket his Christmas present: Bucky Barnes’ vibranium arm.