The correct options are
B once
C after
From the given sentences, one of them ("You'd like it after her") is dependent on the other ("Your friend is reading the newspaper"). Hence, the sentence that's dependent is subordinate, and we need a subordinate conjunction to link the two. The person asking the question and one can read the newspaper only after the friend has finished reading it hence the events would take place one after the other, and the conjunctions 'after' and 'once' both can be used. 'After' means 'during a period of time following an event' and 'once' means 'as soon as', both are relevant to the combined question asked by the speaker, hence options B and C are correct. 'Until' means 'up to the time till', whereas the event in "You'd like it after her" can take place only after the event in the principal sentence "Your friend is reading the newspaper" has taken place, and not till the time that event occurs, hence option A is incorrect. 'While' is used when both the events are taking place at the same time since they're not taking place simultaneously, option D is incorrect.