Most of this has been mentioned in replies and comments, but here is my take.
The conquered have a good chance of learning the basics of the language just by being surrounded by their masters' native language. They'll learn "no" really quickly as they learn that various phrases and terms are perjoratives. Eventually, more and more things will "click" until the slaves can understand enough to work unattended. If a slave has to ask for something to get it and are given an example (like water rations in a previous answer), the slave has a huge incentive to learn. A slave will probably make sure his or her relatives, friends, associates, and former worst enemies know how to say the basics if the conquerors are harsh enough to let people starve, dehydrate, or soil themselves and get a beating before providing basic necessities.
The children--or more likely grandchildren--will have a much easier time. They've grown up with the language. Even if their parents only conversed at a substance level (eat, bathroom, sick, yes, no), their children (assuming the conquerors have this long to assimilate the conquered) will have a better base in their childhood which will facilitate better vocabulary and syntax if they hear or are involved in conversations with fluent speakers. They'll probably have a much easier time if they do the same work as their parents.
Do the conquerors have any sort of religion? Maybe, the prophecies are only preached in the slaves' new language and, in an effort to secure the slaves' salvation (and obedience?), a sect is authorized to teach them to speak and listen (and afterwards get a meal--and, for vetted adherents--an afternoon off) by spreading the good word. In addition to learning the language, religion could serve as a path to becoming an aristocrat's servant. Or, you get the chants right or you're among next week's sacrifices.
There will be exceptional slaves and exceptional slaveowners. If an aristocrat that values aesthetics and art purchases a slave solely for her beauty and finds out she has perfect pitch, the aristocrat will probably try to increase the value (and his enjoyment) of his investment by having a native expert teach her the songs of his people. A laborer might show his foreman an invention or process change with even limited language skills; if the foreman and the slave's owner truly want to integrate their conquered, more effort might be shown to teach the language to the slave. Talent could lead to enfranchisement. If you decide to allow dedicated schooling, perhaps owners might be rewarded for supplying exceptional students with laurels or just free tuition for their chattel. Magna cum laude might mean eventual manumission in a particularly unified government and citizenry.