In recent years, mathematical models based on reaction–diffusion equations with non-local terms have significantly advanced our theoretical understanding of the mechanisms driving the spatial spread and phenotypic evolution of cell populations. In this work, we present a reaction–diffusion partial integro-differential equation (PIDE) model to study tumour growth dynamics, focusing on a continuously varying phenotypic trait that influences cellular responses to oxygen availability. The model couples the dynamics of the tumour cell population with a reaction–diffusion equation governing oxygen concentration, a key factor in cell proliferation. Both equations incorporate spatial anisotropy by means of anisotropic diffusion tensors. Through asymptotic analysis and numerical simulations, we explore travelling wave solutions and show that phenotypic heterogeneity gives rise to complex wave profiles, with different phenotypic states dominating in specific spatial regions of the advancing front. Moreover, we perform three-dimensional simulations of tumour progression in anatomically accurate brain geometries reconstructed from magnetic resonance imaging. These simulations integrate patient-specific oxygen diffusion profiles and preferential cell migration along white matter tracts, as inferred from diffusion tensor imaging (DTI) data.