Nuclear charge radii of 55,56Ni were measured by collinear laser spectroscopy. The obtained information completes the behavior of the charge radii at the shell closure of the doubly magic nucleus 56Ni. The trend of charge radii across the shell closures in calcium and nickel is surprisingly similar despite the fact that the 56Ni core is supposed to be much softer than the 48Ca core. The very low magnetic moment μ(55Ni)=−1.108(20) μN indicates the impact of M1 excitations between spin-orbit partners across the N,Z=28 shell gaps. Our charge-radii results are compared to ab initio and nuclear density functional theory calculations, showing good agreement within theoretical uncertainties.