1) past tense of will
or 2) "conditional" form (i.e., the event is not expected to happen)
both forms are NOT applicable here. for 2), we don't have reasons to suspect/expect that mother's putting of the wallpaper is some hypothetical event.
so when you say "By the time the children get home," you need to indicate a future perfect tense: "will have + past participle"
this signals that an action/event (i.e., mother's putting of wallpaper) will be completed sometime in the future BEFORE another future event (by the time children get home).
think of T = 0 present
T = 1 (future) is mother's putting of wallpaper
and T = 2 (even more distant future) is children get home
hope that helps!
kudos if you like me (or my post)