This paper presents a solution for the state estimation and control problems for a class of unconventional vertical takeoff and landing (VTOL) UAVs operating in forward-flight conditions. A tightly-coupled state estimation approach is used to estimate the aircraft navigation states, sensor biases, and the wind velocity. State estimation is done within a matrix Lie group framework using the Invariant Extended Kalman Filter (IEKF), which offers several advantages compared to standard multiplicative EKFs traditionally used in aerospace and robotics problems. An SO(3)- based attitude controller is employed, leading to a single attitude control law without a separate sideslip control loop. A control allocator is used to determine how to use multiple, possibly redundant, actuators to produce the desired control moments. The wind velocity estimates are used in the attitude controller and the control allocator to improve performance. A numerical example is considered using a sample VTOL tailsitter-type UAV with four control surfaces. Monte-Carlo simulations demonstrate robustness of the proposed control and estimation scheme to various initial conditions, noise levels, and flight trajectories.