The dynamics of two massive black holes in a rotationally supported nuclear disc of mass Mdisc = 108Msolar is explored using N-body/smoothed particle hydrodynamics simulations. Gas and star particles are copresent in the disc. Described by a Mestel profile, the disc has a vertical support provided by turbulence of the gas, and by stellar velocity dispersion. A primary black hole of mass 4 × 106Msolar is placed at the centre of the disc, while a secondary black hole is set initially on an eccentric corotating orbit in the disc plane. Its mass is in a 1:1, 1:4, and 1:10 ratio, relative to the primary. With this choice, we mimic the dynamics of black hole pairs released in the nuclear region at the end of a gas-rich galaxy merger. It is found that, under the action of dynamical friction, the two black holes form a close binary in ~10 Myr. The inspiral process is insensitive to the mass fraction in stars and gas present in the disc and is accompanied by the circularization of the orbit. We detail the gaseous mass profile bound to each black hole that can lead to the formation of two small Keplerian discs, weighing ~2per cent of the black hole mass, and of size ~0.01 pc. The mass of the tightly (loosely) bound particles increases (decreases) with time as the black holes spiral into closer and closer orbits. Double active galactic nucleus activity is expected to occur on an estimated time-scale of <~10 Myr, comparable to the inspiral time-scale. The double nuclear point-like sources that may appear during dynamical evolution will have typical separations of <~10 pc.