The convective heat transfer from two tandemly arranged circular cylinders, with the downstream one subjected to controlled transverse oscillation, is numerically studied in this work. At Reynolds number 100, eight oscillation frequencies varying from 0.4 to 1.7 and two oscillation amplitudes of 0.15 and 0.35 are examined at three spacing distances, which are corresponding to the Vortex Suppression, critical and the Vortex Formation regimes. Results show that in such a system, the spacing distance determines the flow regime and set the base for the flow patterns, the forcing and heat transfer characteristics. For the flow features, the locked-on phenomenon is observed in the present study, whereby the flow and thermal fields are phase-locked with the cylinder oscillation. Outside the locked-on region, an ordered and modulated near-wake can also be observed, but the variation period of the flow field is different from the oscillation period. Additionally, it is found that the parameter range to obtain a locked-on response is narrowed with an increasing spacing distance. Compared with the system with two stationary cylinder, the lift and drag force can subject to larger variations (especially for the downstream cylinder) due to the oscillatory motion. More importantly, the present study demonstrates a large room to enhance the heat transfer. It shows that in the Vortex Suppression and the Vortex Formation regimes considered here, the heat transfer rate of the downstream cylinder can be improved in most cases (with that of the upstream cylinder slightly increased or kept the same with the baseline). A maximum increase of 59.4% of the average Nusselt number can be obtained in the test matrix studied.