Low Earth orbit (LEO) satellites are a crucial component of the future non-terrestrial networks (NTN) due to lower latency, robust signal strengths, shorter revisit times, and dense constellations. However, acquiring reliable channel state information (CSI) in LEO satellite communication remains challenging owing to severe signal attenuation over long propagation distances and short coherence times. Despite these challenges, LEO channels benefit from pronounced line-of-sight dominance and geometric properties inherently tied to positioning information. In this work, we propose an integrated positioning and communication (IPAC) framework for multi-LEO satellite networks to address the unique challenges posed by LEO channels. Specifically, we leverage in-the-loop LEO positioning to exploit users' position information for improving uplink CSI acquisition. To overcome the link-budget limitations of single-satellite systems, cooperative multi-LEO uplink data detection is adopted. By exploiting the different coherent timescales of position-related parameters and random channel gains, we develop a dual-timescale Kalman filter-based IPAC framework: an unscented Kalman filter (UKF) for tracking users' position and velocity in the large-timescale, and a Kalman filter that leverages the position information obtained in the large-timescale for improved data-aided uplink channel estimation in the small-timescale. Finally, the two tasks of channel estimation and cooperative data detection are jointly addressed through the expectation maximization (EM) algorithm. Numerical results demonstrate that the proposed IPAC approach outperforms the conventional baseline in terms of channel estimation accuracy and communication performance.