We are going to calculate the Hawking thermal flux and the flux of energy-momentum tensor of the rotating regular black hole with the time-delay proposed in arXiv:1510.08828, based on the anomaly cancellation. To this purpose, we first try to reduce the field theories to the infinite two-dimensional massless free models in which the anomaly cancellation method is possible, in the three metrics in arXiv:1510.08828. As a result, we demonstrate that the two of them can reduce. Hence we perform the calculation in these two metrics, and finally we obtain the flow of the angular momentum and flux of the Hawking radiation in these two metrics which involves the three effects:~the quantum gravity effect regularizing the gravity sources of the black holes, rotating, and the time-delay. Our result could be considered to correspond to a more realistic Hawking radiations.