ABSTRACT Disentangling the radio flux contribution from star formation (SF) and active-galactic-nuclei (AGNs) activity is a long-standing problem in extragalactic astronomy, since at frequencies of ≲ 10 GHz, both processes emit synchrotron radiation. We present in this work the general objectives of the PARADIGM (PAnchromatic high-Resolution Analysis of DIstant Galaxy Mergers) project, a multi-instrument concept to explore SF and mass assembly of galaxies. We introduce two novel general approaches for a detailed multiscale study of the radio emission in local (ultra) luminous infrared galaxies (U/LIRGs). In this work, we use archival interferometric data from the Very Large Array (VLA) centred at ∼ 6 GHz (C band) and present new observations from the e-Multi-Element Radio-Linked Interferometer Network (e-MERLIN) for UGC 5101, VV 705, VV 250, and UGC 8696. Using our image decomposition methods, we robustly disentangle the radio emission into distinct components by combining information from the two interferometric arrays. We use e-MERLIN as a probe of the core-compact radio emission (AGN or starburst) at ∼ 20 pc scales, and as a probe of nuclear diffuse emission, at scales ∼100–200 pc. With VLA, we characterize the source morphology and the flux density on scales from ∼200 pc up to and above 1 kpc. As a result, we find deconvolved and convolved sizes for nuclear regions from ∼10 to ∼200 pc. At larger scales, we find sizes of 1.5–2 kpc for diffuse structures (with effective sizes of ∼ 300–400 pc). We demonstrate that the radio emission from nuclear extended structures (∼ 100 pc) can dominate over core-compact components, providing a significant fraction of the total multiscale SF output. We establish a multiscale radio tracer for SF by combining information from different instruments. Consequently, this work sets a starting point to potentially correct for overestimations of AGN fractions and underestimates of SF activity.