We present interferometric observations of a multitude of CO lines [12CO(1-0), 12CO(2-1), 13CO(1-0) and 13CO(2-1)] and dense gas tracers [HCN(1-0), HCO+(1-0), HNC(1-0) and HNCO(4-3)] in two nearby edge-on lenticular galaxies, NGC 4710 and NGC 5866. The position-velocity diagrams of both galaxies are X-shaped, revealing that the galaxies are barred, with most of the gas concentrated in a nuclear disc and an inner ring in each galaxy. Relatively tenuous (CO) molecular gas is present in both kinematic components, while dense gas is generally only detected in the nuclear discs. We probe the physical conditions of a two-component molecular interstellar medium in each galaxy and each kinematic component by using molecular line ratio diagnostics in three complementary ways. First, we measure the ratios of the position-velocity diagrams of different lines, second we measure the ratios of the integrated line intensities as a function of projected position in each kinematic component taken separately, and third we model the line ratios using a non-local thermodynamic equilibrium radiative transfer code characterising the kinetic temperature, volume density and column density of the molecular gas. Overall, the nuclear discs appear to have a tenuous molecular gas component that is hotter and optically thinner than that in the inner rings, with a larger dense gas fraction, suggesting more dense clumps immersed in a hotter more diffuse molecular medium. This is consistent with evidence that the physical conditions in the nuclear discs are similar to those in photo-dissociation regions, with intense UV radiation from young stars and few cosmic rays. A similar picture emerges when comparing the observed molecular line ratios with those of other galaxy types and spatially-resolved giant molecular clouds. While there are a number of similarities and differences, the physical conditions of the molecular gas in the nuclear discs of NGC 4710 and NGC 5866 appear intermediate between those of spiral galaxies and starbursts, with intense but not extreme star-formation activity, while the star formation in the inner rings is even milder.