This paper addresses the problem of fast transient simulation of high-speed coupled channels driven by transceivers that include Transmitter Feed-Forward Equalization (TX-FFE). Practical circuit hardware implementations of such drivers may not be correctly represented by industry standard algorithmic modeling interfaces, which are usually based on ideal digital finite impulse response filters. In particular, slow transients may arise when aggressive FFE is activated and particular switching sequences significantly stress local voltage regulators integrated in the output buffers. In this paper, we model this effect using a constructive hierarchical approximation of the analog waveforms that form a linearized vector Thevenin or Norton representation of the driver. This representation enables the accurate simulation of coupled channels including both differential and common-mode signals. Analog waveforms at the receiver input are readily obtained by standard time-frequency transformations, while including channel characteristics and any additional linear equalization block, if present. This proposed framework is compatible with industry-standard Algorithmic Modeling Interfaces (AMI), extending their scope to coupled channels with non-ideal TX-FFE circuit blocks. Eye diagrams and transient waveforms corresponding to millions of bits are computed in few seconds using a non-optimized software implementation. Random jitter and crosstalk are seemlessly included as in standard AMI frameworks. The results of various numerical experiments on commercial transceivers are reported in this paper, confirming both the accuracy and the efficiency of the proposed framework.