MERLIN observations of the 22-GHz H2O masers in the circumstellar envelope of the supergiant NML Cyg show an irregular ring of emission 200 mas across, plus a pair of outlying features to the north-west (NW) and south-east (SE) nearly 600 mas apart. Comparison with maps at previous epochs reveals proper motions of these outlying features, of 1.9 ± 0.4 mas yr-1. This corresponds to a transverse velocity of 19 ± 4 km s-1 at a distance of 2 kpc, which is consistent with the radial velocity range of the 22-GHz emission. The remaining masers which form the irregular ring are modelled as an expanding and accelerating thick shell. The NW-SE symmetry axis of the H2O masers is aligned with a NW-SE asymmetry which occurs in the OH 1612-MHz masers on a larger scale of 3 arcsec. It appears that this is an internally driven effect, most likely a bipolar outflow which is more highly collimated near the star (H2O masers) and becomes less collimated at greater distances (OH masers). The results are compared with bipolar outflows seen in other evolved objects.