"As Dobby clearly states, house-elves are enslaved to families, not individuals."
I am not sure I agree. House elves obey the family members, sure, but it seems the elf has a master he belongs to. This person's orders take precedence. They obey the other family members as long as it doesn't contradict a direct order by their owner. For example in the Crouch family, Winky served Bartemius Crouch Senior. On his orders, she was the keeper of his son, Barty Crouch Junior, and used her magic to subdue him:
"Winky was afraid to see me so angry. She used her own brand of
magic to bind me to her. She pulled me from the tent, pulled me into
the forest, away from the Death Eaters. I tried to hold her back. I
wanted to return to the campsite. I wanted to show those Death
Eaters what loyalty to the Dark Lord meant, and to punish them for
their lack of it. I used the stolen wand to cast the Dark Mark into
the sky. “Ministry wizards arrived. They shot Stunning Spells
everywhere. One of the spells came through the trees where Winky and I
stood. The bond connecting us was broken. ~Harry Potter and the Goblet of Fire - Veritaserum
Barty's story about his escape proves that he could not order Winky to let him go free, otherwise he would have done so. Winky was under instructions from the head of the family to keep him prisoner.
There is also the example when Sirius, head of the Black family, left all his property (including Kreacher) to Harry Potter, who wasn't member of the Black family, and Harry became the owner or Kreacher:
“Well, that simplifies matters,” said Dumbledore cheerfully. “It seems
that Sirius knew what he was doing. You are the rightful owner of
number twelve, Grimmauld Place and of Kreacher.” ~Harry Potter and the Half Blood Prince - Will and Won't
And that's the resolution of Dumbledore's dilemma what to do with an elf whose head is full of secrets of the Order. When Sirius left all his property to Harry, Kreacher became Harry's slave, had to obey Harry's orders and not anyone else's. He may call Bellatrix or Narcissa 'mistress' and Draco 'young master' but those are empty words. Harry is the one he has to obey.
"Mrs Black never thought to order Kreacher to tell her what happened to Regulus, but she could've done. In which case, Kreacher would've been in a real quandary."
It is unclear whom Kreacher had to obey above others in the Black family. The pureblood inheritance seems to follow the principle of primogeniture, and it might be that Regulus' orders took precedence over Walburga's. It also might be that at same point Kreacher was given to Regulus as his personal slave.
Or maybe the precedence was by senority, and had Walburga given a direct order, Kreacher would have told her what happened with Regulus.
In any case, when given a direct order by the owner, or the family members, the elf tries to obey according to his best ability. But there are slips:
“The Mudblood touched Kreacher, he will not allow it, what would his
Mistress say?” “I told you not to call her ‘Mudblood’!” snarled Harry,
but the elf was already punishing himself: He fell to the ground and
banged his forehead on the floor.~Harry Potter and the Deathly Hallows-Kreacher's Tale
And situations when the elf is unable to carry out the order:
“Nothing Kreacher did made any mark upon it,” moaned the elf.
“Kreacher tried everything, everything he knew, but nothing, nothing
would work. . . . So many powerful spells upon the casing, Kreacher
was sure the way to destroy it was to get inside it, but it would not
open. . . . Kreacher punished himself, he tried again, he punished
himself, he tried again. Kreacher failed to obey orders, Kreacher
could not destroy the locket!
That's probably the solution to the elf's dilemma if he receives conflicting orders from his owner or from two or more members of the family - if the elf can't figure out which orders he should follow and is unable to ask for clarifications from his master, as @Rand al'Thor suggests, the elf would probably punish himself for failing to obey orders.