In settings where users both need high accuracy and are time-pressured, such as doctors working in emergency rooms, we want to provide AI assistance that both increases decision accuracy and reduces decision-making time. Current literature focusses on how users interact with AI assistance when there is no time pressure, finding that different AI assistances have different benefits: some can reduce time taken while increasing overreliance on AI, while others do the opposite. The precise benefit can depend on both the user and task. In time-pressured scenarios, adapting when we show AI assistance is especially important: relying on the AI assistance can save time, and can therefore be beneficial when the AI is likely to be right. We would ideally adapt what AI assistance we show depending on various properties (of the task and of the user) in order to best trade off accuracy and time. We introduce a study where users have to answer a series of logic puzzles. We find that time pressure affects how users use different AI assistances, making some assistances more beneficial than others when compared to no-time-pressure settings. We also find that a user's overreliance rate is a key predictor of their behaviour: overreliers and not-overreliers use different AI assistance types differently. We find marginal correlations between a user's overreliance rate (which is related to the user's trust in AI recommendations) and their personality traits (Big Five Personality traits). Overall, our work suggests that AI assistances have different accuracy-time tradeoffs when people are under time pressure compared to no time pressure, and we explore how we might adapt AI assistances in this setting.