The title of the story is somewhat ironic. The first hint is that “custody” usually refers to children, not pumpkins. In the story, Lord Emsworth seems to care more about his prize-contending pumpkin than his own son, whom he has “little use for” (p. 44).
The “fluffy-minded” Earl of Elmsworth has trouble prioritizing (p. 43). He seems to care about his garden and his prize pumpkins, and he is more concerned about his gardener quitting than his son’s desire to marry the gardener’s cousin.
Reason, so violently expelled, came stealing timidly back to her throne, and a cold hand seemed suddenly placed upon his heart. (p. 49)
Uh oh! There’s no one to take care of the pumpkin. The pumpkin will definitely not win any prizes under the care of the “deputy head gardener” who is “not up to the job of preparing his precious pumpkin, "The Hope of Blandings", for the Shrewsbury Show” (enotes Wikipedia page). This is ironic because he is concerned about the pumpkin, not the son.
Page numbers refer to: http://www.unz.org/Pub/WagnallFunk-1927v10-00043