The bulk modulus, K, quantifies the elastic response of an object to an isotropic compression. For soft compressible colloids, knowing K is essential to accurately predict the suspension response to crowding. Most colloids have complex architectures characterized by different softness, which, additionally, depends on compression. In this article, we determine the different values of K for the various morphological parts of individual nanogels and probe the changes of K with compression. Our method uses a partially-deuterated polymer, which exerts the required isotropic stress, and small-angle neutron scattering with contrast matching to determine the form factor of the particles without any scattering contribution from the polymer. We show a clear difference in softness, compressibility and evolution of $K$ between the shell of the nanogel, and the rest of the particle, depending on the amount of crosslinker used in their synthesis.