Does sound like a water flow rate problem.When you first open the tap you are on standing pressure. Once the tap is open it's running pressure which will gradually slow down if there are any restrictions like the main stop tap not fully opened.
Sometimes people fit ball-o-fix valves to the pipework when changing baths e.t.c so they can still use the water in the rest of the house. These valves have quite a narrow bore and can cause a restriction.
Can't really tell much more, but if you have checked every thing is fully opened and no restrictions you might want to ask the water supplier wherever you sre to test your supply.
You can test in a simple way by opening the cold tap upstairs, if that is also slowing down on it's own then it would point to a supply problem...
T.H.S
May 2007