Non-canonical wakes of two-dimensional elliptic cylinders are analysed numerically for their near- and far-wake characteristics. The governing equations are solved using an immersed boundary method based projection scheme. The wakes are then classified into three distinct types according to diverse flow and thermal properties. An unexpected mean temperature evolution along the centreline of the wake is observed for certain wake states. In order to explain this unusual variation, novel heat transport models are constructed based on the vortex dynamics. These models are derived by considering vorticity is acted by flow, which has shear and swirl. Mechanisms of the primary vortex street breakdown and formation of the secondary vortex street are also proposed based on these models. A new phenomenon namely ‘dual near-wall instantaneous recirculation’ is observed, and its appearance is found to be a function of length of the primary von Kármán vortex street. The same phenomenon is also found to be responsible for the secondary peak in the Nusselt number variation along the circumference of the cylinder. Despite varied differences between the wake types, it is observed that the transitions occur through a supercritical Hopf bifurcation in all of them, at least in the von Kármán region of the wake. Low-frequency unsteadiness observed in the far wakes is examined through a signal decomposition method. Our results show that the secondary low frequency is resulting from the transition region which has a negative instability slope. Finally, onset of the primary vortex street breakdown and its scale in terms of Reynolds number is computed.