We present a mathematical study of the emergence of phenotypic heterogeneity in vascularised tumours. Our study is based on formal asymptotic analysis and numerical simulations of a system of non-local parabolic equations that describes the phenotypic evolution of tumour cells and their nonlinear dynamic interactions with the oxygen, which is released from the intratumoural vascular network. Numerical simulations are carried out both in the case of arbitrary distributions of intratumour blood vessels and in the case where the intratumoural vascular network is reconstructed from clinical images obtained using dynamic optical coherence tomography. The results obtained support a more in-depth theoretical understanding of the ecoevolutionary process which underpins the emergence of phenotypic heterogeneity in vascularised tumours. In particular, our results offer a theoretical basis for empirical evidence indicating that the phenotypic properties of cancer cells in vascularised tumours vary with the distance from the blood vessels, and establish a relation between the degree of tumour tissue vascularisation and the level of intratumour phenotypic heterogeneity.