A family of completely integrable nonlinear deformations of systems of N harmonic oscillators are constructed from the non-standard quantum deformation of the sl(2,) algebra. Explicit expressions for all the associated integrals of motion are given and the long-range nature of the interactions introduced by the deformation is shown to be linked to the underlying co-algebra structure. Separability and superintegrability properties of such systems are analysed, and their connection with classical angular momentum chains is used to construct a non-standard integrable deformation of the XXX hyperbolic Gaudin system.