Lay summarisation aims to produce summaries of scientific articles that are comprehensible to non-expert audiences. However, previous work assumes a one-size-fits-all approach, where the content and style of the produced summary are entirely dependent on the data used to train the model. In practice, audiences with different levels of expertise will have specific needs, impacting what content should appear in a lay summary and how it should be presented. Aiming to address this, we propose ATLAS, a novel abstractive summarisation approach that can control various properties that contribute to the overall "layness" of the generated summary using targeted control attributes. We evaluate ATLAS on a combination of biomedical lay summarisation datasets, where it outperforms state-of-the-art baselines using mainstream summarisation metrics. Additional analyses provided on the discriminatory power and emergent influence of our selected controllable attributes further attest to the effectiveness of our approach.