In this line (the last line of the poem) the word "whose" means "belonging to who." In other words, the speaker is telling the listener not to wonder which dead man's sweetheart he (the speaker) is now "cheering."
The point of that last line is that the dead are dead and the living have to continue to live their own lives. We should not be concerned about who the sweetheart used to be in love with. Her old lover is dead and she can't just stop living herself. She must go ahead and continue to live -- to find another love. So don't worry about it when the dead man's lover takes another man...
A.E.Housman's poem "Is My Team Ploughing?" is an imaginary conversation between the young farmer who is now dead and the poet narrator. The dead 'ploughman' ask a series of questions to the poet narrator which are answered patiently by him. The gist of the answers by the poet narrator are that the death of the ploughman has made no difference and life goes on as usual:
No change though you lie under
The land you used to plough
Finally, the dead man is anxious to know whether his lover is still mourning for him. The poet narrator, assures him that it is he who is now caring and comforting her.
"Whose" in the last line obviously refers to the former lover of the dead man. The poet narrator doesn't want to reveal to the dead man the truth that it is he who has become her lover because he does not want to make him jealous.
The message of the poem is that life continues to go on irrespective of some one's death. Lovers's vows and marriage oaths are null and void once a partner dies.