Biomembrane interfaces create regions of slowed water dynamics in their vicinity. When two lipid bilayers come together, this effect is further accentuated, and the associated slowdown can affect the dynamics of larger-scale processes such as membrane fusion. We have used molecular dynamics simulations to examine how lipid and water dynamics are affected as two lipid bilayers approach each other. These two interacting fluid systems, lipid and water, both slow and become coupled when the lipid membranes are separated by a thin water layer. We show in particular that the water dynamics become glassy, and diffusion of lipids in the apposed leaflets becomes coupled across the water layer, while the outer leaflets remain unaffected. This dynamic coupling between bilayers appears mediated by lipid-water-lipid hydrogen bonding, as it occurs at bilayer separations where water-lipid hydrogen bonds become more common than water-water hydrogen bonds. We further show that such coupling occurs in simulations of vesicle-vesicle fusion prior to the fusion event itself. Such altered dynamics at membrane-membrane interfaces may both stabilize the interfacial contact and slow fusion stalk formation within the interface region.