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Recent advances to hardware integration and realization of highly-efficient Compressive Sensing (CS) approaches have inspired novel circuit and architectural-level approaches. These embrace the challenge to design more optimal nonuniform CS solutions that consider device-level constraints for IoT applications wherein lifetime energy, device area, and manufacturing costs are highly-constrained, but meanwhile the sensing environment is rapidly changing. In this manuscript, we develop a novel adaptive hardware-based approach for non-uniform compressive sampling of sparse and time-varying signals. The proposed Adaptive Sampling of Sparse IoT signals via STochastic-oscillators (ASSIST) approach intelligently generates the CS measurement matrix by distributing the sensing energy among coefficients by considering the signal characteristics such as sparsity rate and noise level obtained in the previous time step. In our proposed approach, Magnetic Random Access Memory (MRAM)-based stochastic oscillators are utilized to generate the random bitstreams used in the CS measurement matrix. SPICE and MATLAB circuit-algorithm simulation results indicate that ASSIST efficiently achieves the desired non-uniform recovery of the original signals with varying sparsity rates and noise levels.