We use ALMA and JVLA observations of the galaxy cluster Cl J1449+0856 at z=1.99, in order to study how dust-obscured star-formation, ISM content and AGN activity are linked to environment and galaxy interactions during the crucial phase of high-z cluster assembly. We present detections of multiple transitions of 12 CO, as well as dust continuum emission detections from 11 galaxies in the core of Cl J1449+0856. We measure the gas excitation properties, star-formation rates, gas consumption timescales and gas-to-stellar mass ratios for the galaxies.We find evidence for a large fraction of galaxies with highly-excited molecular gas, contributing >50% to the total SFR in the cluster core. We compare these results with expectations for field galaxies, and conclude that environmental influences have strongly enhanced the fraction of excited galaxies in this cluster. We find a dearth of molecular gas in the galaxies' gas reservoirs, implying a high star-formation efficiency (SFE) in the cluster core, and find short gas depletion timescales τ<0.1-0.4 Gyrs for all galaxies. Interestingly, we do not see evidence for increased specific star-formation rates (sSFRs) in the cluster galaxies, despite their high SFEs and gas excitations. We find evidence for a large number of mergers in the cluster core, contributing a large fraction of the core's total star-formation compared with expectations in the field. We conclude that the environmental impact on the galaxy excitations is linked to the high rate of galaxy mergers, interactions and active galactic nuclei in the cluster core.