Nano-quadcopters are versatile platforms attracting the interest of both academia and industry. Their tiny form factor, i.e., $\,$10 cm diameter, makes them particularly useful in narrow scenarios and harmless in human proximity. However, these advantages come at the price of ultra-constrained onboard computational and sensorial resources for autonomous operations. This work addresses the task of estimating human pose aboard nano-drones by fusing depth and images in a novel CNN exclusively trained in simulation yet capable of robust predictions in the real world. We extend a commercial off-the-shelf (COTS) Crazyflie nano-drone -- equipped with a 320$\times$240 px camera and an ultra-low-power System-on-Chip -- with a novel multi-zone (8$\times$8) depth sensor. We design and compare different deep-learning models that fuse depth and image inputs. Our models are trained exclusively on simulated data for both inputs, and transfer well to the real world: field testing shows an improvement of 58% and 51% of our depth+camera system w.r.t. a camera-only State-of-the-Art baseline on the horizontal and angular mean pose errors, respectively. Our prototype is based on COTS components, which facilitates reproducibility and adoption of this novel class of systems.