In this work, we investigate the challenging problem of on-demand semantic communication over heterogeneous wireless networks. We propose a fidelity-adjustable semantic transmission framework (FAST) that empowers wireless devices to send data efficiently under different application scenarios and resource conditions. To this end, we first design a dynamic sub-model training scheme to learn the flexible semantic model, which enables edge devices to customize the transmission fidelity with different widths of the semantic model. After that, we focus on the FAST optimization problem to minimize the system energy consumption with latency and fidelity constraints. Following that, the optimal transmission strategies including the scaling factor of the semantic model, computing frequency, and transmitting power are derived for the devices. Experiment results indicate that, when compared to the baseline transmission schemes, the proposed framework can reduce up to one order of magnitude of the system energy consumption and data size for maintaining reasonable data fidelity.