In this paper, a deep learning approach is proposed to accurately position wheeled vehicles in Global Navigation Satellite Systems (GNSS) deprived environments. In the absence of GNSS signals, information on the speed of the wheels of a vehicle (or other robots alike), recorded from the wheel encoder, can be used to provide continuous positioning information for the vehicle, through the integration of the vehicle's linear velocity to displacement. However, the displacement estimation from the wheel speed measurements are characterised by uncertainties, which could be manifested as wheel slips or/and changes to the tyre size or pressure, from wet and muddy road drives or tyres wearing out. As such, we exploit recent advances in deep learning to propose the Wheel Odometry neural Network (WhONet) to learn the uncertainties in the wheel speed measurements needed for correction and accurate positioning. The performance of the proposed WhONet is first evaluated on several challenging driving scenarios, such as on roundabouts, sharp cornering, hard-brake and wet roads (drifts). WhONet's performance is then further and extensively evaluated on longer-term GNSS outage scenarios of 30s, 60s, 120s and 180s duration, respectively over a total distance of 493 km. The experimental results obtained show that the proposed method is able to accurately position the vehicle with up to 93% reduction in the positioning error of its original counterpart after any 180s of travel. WhONet's implementation can be found at https://github.com/onyekpeu/WhONet.