Good thing Bucky was more or less banking on this.

He meets Sam at the stables, more than aware of the eyes on them. People who care about their Prince and wish to see him thrive. People who know of Bucky's own history in the war.

Greets the Prince with a bow and a lopsided smile. Rides at his side until they are beyond the palace gates. "The witch asked me to bring you to her." Seems a better opening than 'I can't stop thinking of the way you sound when you are overtaken'. Even if both are equally true.