We provide a quantization of the Schwarzschild spacetime in the presence of a cosmological constant, based on midisuperspace methods developed in the spherically symmetric sector of loop quantum gravity, using in particular the 'improved dynamics' scheme. We include both the de Sitter and anti-de Sitter cases. We find that the quantization puts a Planckian positive upper limit on the possible values of the cosmological constant similar to the bounds obtained earlier from studies of homogeneous spacetimes. This means that, for negative cosmological constant, no negative bound is found. Moreover, using semiclassical physical states, we obtain the effective metric and demonstrate the causal structure for various cases. Quantum gravity modifications ensure that the singularity is replaced by a transition surface in all the cases, where the curvature invariants approach mass-independent Planckian bounds. Analysis of the effective stress-energy tensor shows that the null energy condition is strongly violated in the vicinity of the transition surface. Moreover, it shows a weaker asymptotic fall off for a nonvanishing cosmological constant, which could have interesting phenomenological implications.