We present interferometric observations of CO lines [(CO)-C-12(1-0, 2-1) and (CO)-C-13(1-0, 2-1)] and dense gas tracers [HCN(1-0), HCO+(1-0), HNC(1-0) and HNCO(4-3)] in two nearby edge-on barred lenticular galaxies, NGC 4710 and NGC 5866, with most of the gas concentrated in a nuclear disc and an inner ring in each galaxy. 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; secondly, we measure the ratios of each kinematic component's integrated line intensities as a function of projected position; and thirdly we model these line ratios using a non-local thermodynamic equilibrium radiative transfer code. Overall, the nuclear discs appear to have a tenuous molecular gas component that is hotter, optically thinner and with a larger dense gas fraction than that in the inner rings, 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. A similar picture emerges when comparing the observed molecular line ratios with those of other galaxy types. The physical conditions of the molecular gas in the nuclear discs of NGC 4710 and NGC 5866 thus appear intermediate between those of spiral galaxies and starbursts, while the star formation in their inner rings is even milder.