Motivated by applications in autonomous UAV landings on moving platforms, this paper proposes a Variable Horizon Model Predictive Control (VH-MPC) algorithm for cooperative rendezvous problems. Compared to existing VH-MPC, for which the associated computations are extensive which makes implementation on real-time UAV-platform systems most difficult, the look-ahead horizon in our VH-MPC algorithm adapts to the distance and time left to reach the rendezvous state in a computationally tractable manner. Our main contribution is the derivation of these efficient horizon-update rules. More specifically, the computational concerns in standard MPC for rendezvous maneuvers stem from that for the MPC to find a feasible solution, the look-ahead time needs to be long enough to ensure that a complete trajectory to the target set exists (i.e., the position and point in time where the two agents should meet). However, choosing a too long horizon results in expensive computations. A variable horizon can be used to find a horizon that is just long enough to make the control problem feasible, while reducing the computational complexity as the target set gets closer. To validate our proposed VH-MPC scheme, we conduct several experiments both in a realistic simulation environment (FlightGear-JSBSim, which includes nonlinear and complex dynamical effects), and in outdoors experiments with a quadrotor. Our experiments demonstrate i) the prohibitive computational cost of standard MPC, and ii) successful real-time computations of feasible trajectories and control inputs for an autonomous cooperative landing (fixed-wing UAV landing on an unmanned sea-surface vehicle), while satisfying important spatial safety-constraints (e.g., zones around the landing platform to avoid). Our experiments establish the feasibility of important future real-world applications in, e.g., sea rescue missions with fixed-wing drones and autonomous sea vessels.