We present 20 yr of radial velocity (RV) measurements of the M1 dwarf Gl15A, combining five years of intensive RV monitoring with the HARPS-N spectrograph with 15 yr of archival HIRES/Keck RV data. We have carried out an MCMC-based analysis of the RV time series, inclusive of Gaussian Process (GP) approach to the description of stellar activity induced RV variations. Our analysis confirms the Keplerian nature and refines the orbital solution for the 11.44-day period super Earth, Gl15A b, reducing its amplitude to 1.68-0.18+0.17 m s-1 (M sin i = 3.03-0.44+0.46 M⊕), and successfully models a long-term trend in the combined RV dataset in terms of a Keplerian orbit with a period around 7600 days and an amplitude of 2.5-1.0+1.3 m s-1, corresponding to a super-Neptune mass (M sin i = 36-18+25 M⊕) planetary companion. We also discuss the present orbital configuration of Gl15A planetary system in terms of the possible outcomes of Lidov-Kozai interactions with the wide-separation companion Gl15B in a suite of detailed numerical simulations. In order to improve the results of the dynamical analysis, we have derived a new orbital solution for the binary system, combining our RV measurements with astrometric data from the WDS catalogue. The eccentric Lidov-Kozai analysis shows the strong influence of Gl15B on the Gl15A planetary system, which can produce orbits compatible with the observed configuration for initial inclinations of the planetary system between 75° and 90°, and can also enhance the eccentricity of the outer planet well above the observed value, even resulting in orbital instability, for inclinations around 0° and 15°-30°. The Gl15A system is the multi-planet system closest to Earth, at 3.56 pc, and hosts the longest period RV sub-Jovian mass planet discovered so far. Its orbital architecture constitutes a very important laboratory for the investigation of formation and orbital evolution scenarios for planetary systems in binary stellar systems. Based on observations made with the Italian Telescopio Nazionale Galileo (TNG), operated on the island of La Palma by the INAF - Fundación Galileo Galilei at the Roche de Los Muchachos Observatory of the Instituto de Astrofísica de Canarias (IAC); photometric observations made with the robotic telescope APT2 (within the EXORAP programme) located at Serra La Nave on Mt. Etna.