We report on the first multi-wavelength Swift monitoring campaign performed on SDSS J164100.10+345452.7, a nearby narrow-line Seyfert 1 galaxy that had formerly been considered to be radio-quiet. It has, however, more recently been detected both in the radio (at 37 GHz) and in the γ-ray, a behaviour that hints at the presence of a relativistic jet. During our 20-month Swift campaign, while pursuing the primary goal of assessing the baseline optical/UV and X-ray properties of SDSS J164100.10+345452.7, we observed two radio flaring episodes, namely, one each year. Our strictly simultaneous multi-wavelength data closely match the radio flare and allow us to unambiguously link the jetted radio emission of SDSS J164100.10+345452.7. Indeed, for the X-ray spectra preceding and following the radio flare, a simple absorbed power-law model does not offer an adequate description and, thus, an extra absorption component is required. The average spectrum of SDSS J164100.10+345452.7 can best be described by an absorbed power-law model with a photon index Γ = 1.93 ± 0.12, modified by a partially covering neutral absorber with a covering fraction of f = 0.91_−0.03^+0.02. On the contrary, the X-ray spectrum closest to the radio flare does not require any such extra absorber and it is much harder (Γ_flare ∼ 0.7 ± 0.4), thus implying the emergence of an additional, harder spectral component. We interpret this as the jet emission emerging from a gap in the absorber. The fractional variability we derived in the optical/UV and X-ray bands is found to be lower than the typical values reported in the literature because our observations of SDSS J164100.10+345452.7 are dominated by the source being in a low state, as opposed to the literature, where the observations were generally taken as a follow-up of bright flares in other energy bands. Based on the assumption that the origin of the 37 GHz radio flare is the emergence of a jet from an obscuring screen also observed in the X-ray, the derived total jet power is P_jet^tot = 3.5 × 10⁴² erg s⁻¹. This result is close to the lowest values measured in the literature.