Gamma-ray bursts (GRBs) are potential sources of high-energy (> 100 TeV) neutrinos and ultra-high-energy (> 10^9 GeV) cosmic rays (UHECRs). Recent neutrino searches have constrained the connection between them in the one-zone version of the internal shock model. It calculates the prompt particle emission from a single representative collision of plasma shells in the GRB jet, assuming that the jet parameters inferred from gamma-rays observations are representative for neutrino and cosmic-ray emission. Yet, in the internal shock model, the prompt emission must originate from multiple collision zones. Efficient energy dissipation implies a spread in the position of the collisions. We produce light curves of gamma rays in different wavelength bands and of neutrinos, from the properties of the central GRB engine. We predict a minimal neutrino flux which, contrary to conventional estimates, hardly depends on parameters such as baryonic loading --- the proportion of energy in baryons --- and average Lorentz boost of the jet. We also find interesting relationships between delays in high-energy gamma-ray bands observable by Fermi and CTA, neutrino emission efficiency, and shape features of the gamma-ray light curves which can be traced back to the properties of the central engine.