This paper presents an analysis of the $\Lambda b0\to J/\psi \Lambda$ angular distribution and the transverse production polarisation of $\Lambda b0$ baryons in proton-proton collisions at centre-of-mass energies of 7, 8 and 13 TeV. The measurements are performed using data corresponding to an integrated luminosity of 4.9 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, collected with the LHCb experiment. The polarisation is determined in a fiducial region of $\Lambda b0$ transverse momentum and pseudorapidity of GeV$/c$ and , respectively. The data are consistent with $\Lambda b0$ baryons being produced unpolarised in this region. The parity-violating asymmetry parameter of the $\Lambda \to p\pi -$ decay is also determined from the data and its value is found to be consistent with a recent measurement by the BES,III collaboration.