We report a local-probe investigation of the magnetically anisotropic kagome compound Nd3Ga5SiO14. Our zero-field muon spin relaxation (muSR) results provide direct evidence of a fluctuating collective paramagnetic state down to 60 mK, supported by a wipeout of the Ga nuclear magnetic resonance (NMR) signal below 25 K. At 60 mK a dynamics crossover to a much more static state is observed by muSR in magnetic fields above 0.5 T. Accordingly, the NMR signal is recovered at low T above a threshold field, revealing a rapid temperature and field variation of the magnetic fluctuations.