Sparse tensors are the natural way to store and represent multi-dimensional data, but ensuring their efficient processing is an important open challenge. In this work, novel portable and highly data-parallel software approaches and specialized FPGA designs are proposed for the most prominent sparse tensor methods: Tensor Times Matrix (TTM) and Matricised Tensor Times Khatri-Rao Product (MTTKRP), which allow for their efficient cross-platform SYCL-based processing in modern heterogeneous systems at different granularity levels. We also conduct an in-depth analytical and experimental roofline-based characterization of the processing upper-bounds that these sparse tensor methods can achieve in multi-core CPU, GPU, heterogeneous, and FPGA-based processing platforms from different vendors, achieving speedups of up to \(6\times \) for TTM and \(7\times \) for MTTKRP, when compared to the state-of-the-art approaches.