Some of the key dystopian elements in Never Let Me Go are lack of individualism (with the corollary of lack of choice), the prevalence of illness, and the failure of the educational system.
A dystopia is a society that has deviated from its utopian aspirations, turning the positive features into negative ones. The people who control England in the future, as Kazuo Ishiguro depicts it, see themselves as individuals; each of them makes a choice to have clones created so that they have a supply of body parts to replace theirs when the parts fail. In order to achieve these perfect individual matches, however, each person must duplicate themself and could conceivably create multiple copies. Their very effort to maintain their own individuality has been doomed through this exact replication.
For the clones, the lack of individualism extends to their lack of choice. As their originals do not consider them human, they have no say in what will become of them. The choices that the original humans make about prolonging their lives are not available to the clones, who are utterly at the mercy of those who created them. Even the rumors of options, such as living in couples, turn out to be lies.
The powers that be had decided that they could ameliorate the damage of the clones’ impending demise by raising them with a positive environment, including such features as art education. This system is a strongly dystopian element because it is based on hypocrisy. Art depends on the artist’s internal searching so that they can express important truths about the human condition. For the clones, who have no knowledge of how society actually functions, what passes for art is a travesty of creativity. Their education ultimately consists of learning that everyone has lied to them.