Follow-up observations of (sub-)mm-selected gravitationally lensed systems have allowed a more detailed study of the dust-enshrouded phase of star formation up to very early cosmic times. Here, the case of the gravitationally lensed merger in HATLAS J142935.3-002836 (also known as H1429-0028; z(lens) = 0.218, z(bkg) = 1.027) is revisited following recent developments in the literature and new Atacama Pathfinder EXperiment (APEX) observations targeting two carbon monoxide (CO) rotational transitions J(up) = 3 and 6. We show that the line profiles comprise three distinct velocity components, where the fainter high velocity one is less magnified and more compact. The modelling of the observed spectral line energy distribution of CO J(up) = 2-6 and [C I] P-3(1)-P-3(0) assumes a large velocity gradient scenario, where the analysis is based on four statistical approaches. Since the detected gas and dust emission comes exclusively from only one of the two merging components (the one oriented north-south, NS), we are only able to determine upper limits for the companion. The molecular gas in the NS component in H1429-0028 is found to have a temperature of similar to 70 K, a volume density of log (n[cm(-3)]) similar to 3.7, to be expanding at similar to 10 km s(-1) pc(-1), and amounts to M-H2 = 4(-2)(+3) x 10(9)M(circle dot). The CO to H-2 conversion factor is estimated to be alpha(CO) = 0.4(-0.2)(+0.3) M-circle dot/(K km s(-1) pc(2)). The NS galaxy is expected to have a factor of greater than or similar to 10x more gas than its companion (M-H2 less than or similar to 3 x 10(8) M-circle dot). Nevertheless, the total amount of molecular gas in the system comprises only up to 15 per cent (1 sigma upper limit) of the total (dynamical) mass.