The helicity-dependent single π⁰ photoproduction cross section on the deuteron and the angular dependence of the double polarisation observable E for the quasi-free single π⁰ production off the proton and the neutron have been measured, for the first time, from the threshold region up to the photon energy 1.4 GeV. The experiment was performed at the tagged photon facility of the MAMI accelerator and used a circularly polarised photon beam and longitudinally polarised deuteron target. The reaction products were detected using the large acceptance Crystal Ball/TAPS calorimeter, which covered 97% of the full solid angle. Comparing the cross section from the deuteron with the sum of free nucleon cross sections provides a quantitative estimate of the effects of the nuclear medium on pion production. In contrast, comparison of the E helicity asymmetry data from quasi-free protons off deuterium with data from a free proton target indicates that nuclear effects do not significantly affect this observable. As a consequence, it is deduced that the helicity asymmetry E on a free neutron can be reliably extracted from measurements on a deuteron in quasi-free kinematics.