Born in rapidly evolving mini-haloes during the first billion years of the Universe, supermassive black holes (SMBHs) feed from gas flows spanning many orders of magnitude, from the cosmic web in which they are embedded to their event horizon. As such, accretion on to SMBHs constitutes a formidable challenge to tackle numerically, and currently requires the use of sub-grid models to handle the flow on small, unresolved scales. In this paper, we study the impact of resolution on the accretion pattern of SMBHs initially inserted at the heart of dense galactic gas clouds, using a custom super-Lagrangian refinement scheme to resolve the black hole (BH) gravitational zone of influence. We find that once the self-gravitating gas cloud host is sufficiently well resolved, accretion on to the BH is driven by the cloud internal structure, independently of the BH seed mass, provided dynamical friction is present during the early stages of cloud collapse. For a pristine gas mix of hydrogen and helium, a slim disc develops around the BH on sub-parsec scales, turning the otherwise chaotic BH accretion duty cycle into an episodic one, with potentially important consequences for BH feedback. In the presence of such a nuclear disc, BH mass growth predominantly occurs when infalling dense clumps trigger disc instabilities, fuelling intense albeit short-lived gas accretion episodes.