We present the analysis of 61 nucleated dwarf galaxies in the central regions (<~R_vir_/4) of the Fornax galaxy cluster. The galaxies and their nuclei are studied as part of the Next Generation Fornax Survey using optical imaging obtained with the Dark Energy Camera mounted at Blanco/Cerro Tololo Inter-American Observatory and near-infrared data obtained with VIRCam at VISTA/ESO. We decompose the nucleated dwarfs in nucleus and spheroid, after subtracting the surface brightness profile of the spheroid component and studying the nucleus using point source photometry. In general, nuclei are consistent with colors of confirmed metal-poor globular clusters, but with significantly smaller dispersion than other confirmed compact stellar systems in Fornax. We find a bimodal nucleus mass distribution with peaks located at log(M_*_/M_{sun}_)~5.4 and ~6.3. These two nucleus subpopulations have different stellar population properties: the more massive nuclei are older than ~2Gyr and have metal-poor stellar populations (Z<~0.02Z_{sun}_), while the less massive nuclei are younger than ~2Gyr with metallicities in the range 0.02<Z/Z_{sun}_<=1. We find that the nucleus mass (M_nuc_) versus galaxy mass (M_gal_) relation becomes shallower for less massive galaxies starting around 10^8^M_{sun}_, and the mass ratio {eta}_n_=M_nuc_/M_gal_ shows a clear anticorrelation with M_gal_ for the lowest masses, reaching 10%. We test current theoretical models of nuclear cluster formation and find that they cannot fully reproduce the observed trends. A likely mixture of in situ star formation and star cluster mergers seems to be acting during nucleus growth over cosmic time.