Nitrogen-vacancy (NV) center in diamond is a promising quantum metrology tool finding applications across disciplines. The spin sensor measures magnetic fields, electric fields and temperature with nano-scale precision and is fully operable under ambient conditions. Moreover, it achieves precision scaling inversely with total measurement time σB ∝ 1/T (Heisenberg scaling) rather than as the inverse of the square root of T, with σB = √T the Shot-Noise limit. This scaling can be achieved by means of phase estimation algorithms (PEAs), in combination with single-shot read-out. Despite their accuracy, the range of applicability of PEAs is limited to sensing single frequencies with negligible temporal fluctuations. Nuclear Magnetic Resonance (NMR) signals from molecules often contain multifrequency components and sensing them using PEA is ruled out. Here we propose an alternative method for precision magnetometry in frequency multiplexed signals via compressive sensing (CS) techniques focusing on nanoscale NMR. We show that CS can provide for precision scaling approximately as σB ≈ 1/T, as well as for a 5-fold increase in sensitivity over dynamic-range gain, in addition to reducing the total number of resources required. We illustrate our method by taking model solid-state spectra of Glycine acquired under Magic Angle Spinning conditions.