The integration of cold atomic systems with optical nanofibres (ONFs) is an increasingly important experimental platform. Here, we report on the spectra observed during a strongly driven, single-frequency, two-photon excitation of cold rubidium atoms near an ONF. At resonance, two competitive processes, namely a higher excitation rate and stronger pushing of atoms from the nanofibre due to resonance scattering, need to be considered. We discuss the processes that lead to the observed two-peak profile in the fluorescence spectrum as the excitation laser is scanned across the resonance, noting that the presence of the ONF dramatically changes the fluorescence signal. These observations are useful for experiments where high electric field intensities near an ONF are needed, for example when driving nonlinear processes.