We construct $s$-interleaved linearized Reed--Solomon (ILRS) codes and variants and propose efficient decoding schemes that can correct errors beyond the unique decoding radius in the sum-rank metric. The proposed interpolation-based scheme for ILRS codes can be used as a list decoder or as a probabilistic unique decoder that corrects errors of sum-rank up to $t\leq\frac{s}{s+1}(n-k)$, where $s$ is the interleaving order, $n$ the length and $k$ the dimension of the code. Upper bounds on the list size and the decoding failure probability are given where the latter is based on a novel Loidreau--Overbeck-like decoder for ILRS codes. We show how the proposed decoding schemes can be used to decode errors beyond the unique decoding radius in the skew metric by using an isometry between the sum-rank metric and the skew metric. We generalize fast minimal approximant basis interpolation techniques to obtain efficient decoding schemes for ILRS codes (and variants) with subquadratic complexity in the code length. Up to our knowledge, the presented decoding schemes are the first being able to correct errors beyond the unique decoding region in the sum-rank and skew metric. The performance of the proposed decoding schemes and the tightness of the upper bound on the decoding failure probability are validated via Monte Carlo simulations.