We present the first general relativistic hydrodynamic models of the launch and evolution of relativistic jets and winds, driven by thermal energy deposition, possibly due to neutrino-antineutrino annihilation, in the close vicinity of black hole-accretion torus systems. The latter are considered to be the remnants of compact object mergers. Our two-dimensional simulations establish the link between such mergers and future observations of short gamma-ray bursts (GRBs) by the SWIFT satellite. They show that ultrarelativistic outflow with maximum terminal Lorentz factors (Gamma) around 1000 develops for polar energy deposition rates above some 1e48 erg/s per steradian, provided the merger environment has a sufficiently low baryon density. Due to the collimation by the dense accretion torus the typical semi-opening angles of the Gamma > 100 cone are 5-10 degrees, corresponding to about 0.4-1.5% of the hemisphere and apparent isotropized energies (kinetic plus internal) up to ~1e51 erg. 10-30% of the deposited energy are transferred to the outflow with Gamma > 100. Our models confirm the viability of post-merger BH-torus systems as engines of short, hard GRBs and can explain the durations of all observed short GRBs, because different propagation velocities of the front and rear ends lead to a radial stretching of the ultrarelativistic fireball before transparency is reached. The ultrarelativistic flow reveals a highly non-uniform structure with Lorentz factor variations up to factors of a few, caused by the action of Kelvin-Helmholtz instabilities that originate at the fireball-torus interface (abbreviated).