Recent X-ray and Sunyaev-Zeldovich (SZ) observations have detected thermal emission between early-stage merging galaxy clusters. The main purpose of this work is to investigate the properties of the non-thermal emission in the interacting cluster pairs Abell 0399-Abell 0401 and Abell 21-PSZ2 G114.9. These two unique cluster pairs have been found in an interacting state. In both cases, their connection along a filament is supported by an SZ effect detected by the Planck satellite and, in the special case of Abell 0399-Abell 0401, the presence of a radio bridge has been already confirmed by LOFAR observations at 140 MHz. Here, we analyse new high-sensitivity, wideband (250-500 MHz) uGMRT data of these two systems and describe an injection procedure to place limits on the spectrum of Abell 0399-Abell 0401 and on the radio emission between Abell 21-PSZ2 G114.9. In both cases, the low-surface-brightness diffuse emission is not detected in Band 3 (250-500 MHz). For the A399-A401 pair, we are able to constrain the steep spectral index of the bridge emission to be $ with a 95<!PCT!> confidence level between 140 MHz and 400 MHz. We also detect a small patch of the bridge with a flatter spectral index, which may suggest a variable spectral index distribution across the bridge area. For the A21-PSZ2 G114.9 pair, we are able to place an upper limit on the flux density of the bridge emission with two different methods, finding at the central frequency of 383 MHz a conservative value of u <260$ mJy at a 95<!PCT!> confidence level, and a lower value of $f_ u <125$ mJy at an 80<!PCT!> confidence level, based on visual inspection and a morphological criterion. Our work provides a constraint on the spectrum in the bridge A399-A401 that disfavours shock acceleration as the main mechanism for the radio emission. The methods that we propose for the limits on the radio emission in the A21-PSZ2 G114.9 system represent a first step towards a systematic study of these sources.