Double-network (DN) hydrogels are fabricated from poly(2-ethyl-2-oxazoline) (PEOXA)-peptide conjugates, which can be enzymatically crosslinked in the presence of Sortase A (SA), and physical networks of alginate (Alg), yielding matrices with improved mechanical properties with respect to the corresponding PEOXA and Alg single networks and excellent cell viability of encapsulated human auricular chondrocytes (hACs). The addition of a low content of cellulose nanofibrils (CNFs) within DN hydrogel formulations provides the rheological properties needed for extrusion-based three-dimensional (3D) printing, generating constructs with a good shape fidelity. In the presence of hACs, PEOXA-Alg-CNF prehydrogel mixtures can be bioprinted, finally generating 3D-structured DN hydrogel supports showing a cell viability of more than 90%. Expanding the application of poly(2-alkyl-2-oxazoline)-based formulations in the design of tissue-engineering constructs, this study further demonstrates how SA-mediated enzymatic crosslinking represents a suitable and fully orthogonal method to generate biocompatible hydrogels with fast kinetics.