  Disentangling the radio flux contribution from star formation (SF) and
active-galactic-nuclei (AGN) activity is a long-standing problem in
extragalactic astronomy, since at frequencies of $\lesssim$ 10 GHz, both
processes emit synchrotron radiation. We present in this work the general
objectives of the PARADIGM Project, a multi-instrument concept to explore
star-formation and mass assembly of galaxies. We introduce two novel general
approaches for a detailed multiscale study of the radio emission in local
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 UGC5101,
VV705, VV250 and UGC8696. 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
characterise 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 pc 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 star
formation 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.
