This study shows how nucleosome packaging controls which ubiquitin-proteasome co-factors can reach the tumour-suppressor p53 when it is bound to chromatin: the E3 ligase E6-E6AP cannot engage nucleosome-bound p53, whereas the deubiquitinase USP7 readily does, both in vitro and in cells, with a cryo-EM structure capturing USP7 bound to p53 and the nucleosome. Flow Induced Dispersion Analysis (FIDA) was used as a supporting in-solution method to measure the binding affinities (dissociation constants) of fluorescently labelled p53 to its DNA and nucleosome substrates. It is an example of using FIDA to quantify protein–DNA and protein–nucleosome binding affinity in solution.