In this paper, we analyze the performance of a multitask end-to-end transformer model on the task of conversational recommendations, which aim to provide recommendations based on a user's explicit preferences expressed in dialogue. While previous works in this area adopt complex multi-component approaches where the dialogue management and entity recommendation tasks are handled by separate components, we show that a unified transformer model, based on the T5 text-to-text transformer model, can perform competitively in both recommending relevant items and generating conversation dialogue. We fine-tune our model on the ReDIAL conversational movie recommendation dataset, and create additional training tasks derived from MovieLens (such as the prediction of movie attributes and related movies based on an input movie), in a multitask learning setting. Using a series of probe studies, we demonstrate that the learned knowledge in the additional tasks is transferred to the conversational setting, where each task leads to a 9%-52% increase in its related probe score.