A project has been launched for the development of a laser-based neutron source with the few-cycle lasers available at ELI ALPS. Here we show the first experiments, when deuterons were accelerated from ultrathin deuterated foils at 1 Hz repetition rate with the use of 12 fs, 21 mJ laser pulses. The energy spectra of the accelerated deuterons were measured with Thomson ion spectrometers both in forward and backward directions. The accelerated deuterons induced 2H + 2H fusion reaction in a deuterated polyethylene disk. The resulting fast neutrons were measured with a time-of-flight (ToF) detector system, within which each detector consisted of a plastic scintillator and a photomultiplier, at four different angles relative to the normal of the neutron converter disk. We found good agreement with the simulated angular distribution and energy spectra. Here, we also present preparations for the next phases when the repetition rate is increased to 10 Hz. The developed flat liquid jet was demonstrated to accelerate protons over 0.6 MeV cutoff energy with a stability better than 4% for 15 min. We developed two further neutron measurement techniques: a liquid scintillator, the ToF signal of which was evaluated with the pulse shape discrimination method, and a bubble detector spectrometer calibrated against a conventional PuBe source. One of the first upcoming applications is the irradiation of zebrafish embryos with laser-generated ultrashort bunch neutrons. As this experiment needs to be implemented in vacuum, the steps of careful preparation and calibration measurements are also discussed.