This paper presents an analysis of the Lambda(0)(b) -> J/psi Lambda angular distribution and the transverse production polarisation of Lambda(0)(b) baryons in proton-proton collisions at centre-of-mass energies of 7, 8 and 13TeV. The measurements are performed using data corresponding to an integrated luminosity of 4.9 fb(-1), collected with the LHCb experiment. The polarisation is determined in a fiducial region of Lambda(0)(b) transverse momentum and pseudorapidity of 1 < p(T) < 20 GeV/c and 2 < eta < 5, respectively. The data are consistent with Lambda(0)(b) baryons being produced unpolarised in this region. The parity-violating asymmetry parameter of the Lambda -> p pi(-) decay is also determined from the data and its value is found to be consistent with a recent measurement by the BES III collaboration.