The charity, which maintains the 17th Century riverside property, says Thames Water has not shared all of the details for the plans.
Richmond Council says it is standing unanimously against the abstraction scheme.
Thames Water wants to take water from the Thames at Teddington Weir and divert it to the Lee Valley Reservoirs.
The missing water would be replaced with treated waste-water from Mogden Sewage Treatment Works.
Thames Water says the plan is necessary to address water shortages post 2030.