We present a method for measuring the average temperature of a cloud of cold 85Rb atoms in a magneto-optical trap using an optical nanofibre. A periodic spatial variation is applied to the magnetic fields generated by the trapping coils and this causes the trap centre to oscillate, which, in turn, causes the cloud of cold atoms to oscillate. The optical nanofibre is used to collect the fluorescence emitted by the cold atoms, and the frequency response between the motion of the centre of the oscillating trap and the cloud of atoms is determined. This allows us to make measurements of cloud temperature both above and below the Doppler limit, thereby paving the way for nanofibres to be integrated with ultracold atoms for hybrid quantum devices.