We investigate the formation of wrinkling instabilities at the interface between layers of hydrogel and water, which arise to relieve horizontal compressive stresses caused by either differential swelling or confinement. Modelling the gel using a linear-elastic-nonlinear-swelling approach, we determine both a criterion for marginal stability and the growth rates of normal modes. Furthermore, our formalism allows us to understand the influence of differential swelling on the stability of hydrogels brought into contact with water, and we find three distinct phases of the instability. Initially, when only a thin skin layer of gel has swollen, buckles grow rapidly and the gel deforms as an incompressible material. A balance between normal elastic stress and pore pressure selects a wavelength for these buckles that increases with the square root of time. At late times, when the gel approaches a uniformly swollen state, buckles can only grow by differential swelling on much slower timescales determined by solvent transport. At intermediate times, growth is driven by the same fluid transport process as at late times but gradients in fluid pressure in the gel as it swells destabilize the interface, driving faster growth of wrinkles. We also explain why some instabilities can be transient, "healing" as time progresses, while others must remain for all time.