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We investigate the heat transfer and coherent structures in Taylor–Couette (TC) flows that undergo thermal convection driven by an axially applied temperature gradient. Direct numerical simulations are performed in a Rayleigh number range 
$10^6 \leq Ra \leq 3 \times 10^8$ for Prandtl number 
$Pr = 4.38$ and with the shear Reynolds number up to 
$Re = 10^4$. When the rotation number 
$R_f$ increases, the flows undergo a transition from buoyancy-dominated (
$R_f<1$) to shear-dominated convection (
$R_f>1$). In the buoyancy-dominated regime with weak rotations, the flow features are similar to those in Rayleigh–Bénard (RB) convection with large-scale plumes emanating from the thermal boundary layers. In this regime, the 
$Re$-dependence of heat transport 
$Nu$ is sensitive to 
$Ra$. We find that for low 
$Ra$, 
$Nu$ decreases with increasing 
$Re$ and becomes independent of 
$Re$ at high 
$Ra$. In the shear-dominated regime, the flow structures are characterised by Taylor vortices (TVs), which effectively enhance the heat transport. With sufficiently high Reynolds number for 
$2000< Re \le 10\,000$, the flow structures are dominated by turbulent TVs, and the transport scaling laws of heat and angular velocity fluxes become independent of buoyancy. We report that in this turbulent regime the axial heat-transport scaling 
$(Nu\sim Re^{0.578\pm 0.018})$ is consistent with the scaling of radial angular-momentum transport 
$(Nu_{\omega }\sim Re^{0.581\pm 0.026})$.
