A lot of people think Bucky was drafted or have other reasons why Bucky enlisted but think for a second. It was world war two Hitler was at the height of his reign and was having people killed left and right for being Jewish gay and especially sick. As sick as Steve or sicker. If the axis powers had won the war Steve would be amoungst the first to die. He may have had the look to be spared but any of his ailments could have gotten him killed.
Bucky has been there to protect him since they were kids. Whether it was just some back alley right to a simple cold that could kill him. Enlisting in this war was the ultimate way for how Bucky could protect Steve. Becuase even if he died but they won the war Steve would be safe. And if they lost he could always say he did his best then high tail it home to protect steve by his side instead of the front line.
So when Steve saves him and brought him and the others back and after let’s hear it for Captain America scene. Bucky realizes that Steve doesn’t need him anymore and that hurts him. But I’m the bar where Steve asks him is he ready to join Captain America. He just smiles realizing that Steve will always need him. Hell he was even grilling peggy to see her intentions with Steve and would see if she would jump over to him if Steve said no. (Also lets face it he was very jealous too)
Even when he picked up Steve’s sheild at the end his first instinct is to protect Steve.