You're right in thinking that this problem is to do with the tap washer... a small hard rubber disk that lives inside the body of the tap. There are two main possibilities:
1 The washer has perished and swelled in size so that it is restricting the flow, even when the tap is full on.
or, more likely,
2 The retaining screw holding the washer in place inside the tap has come loose or snapped off so that the washer moves about inside the tap body giving this variable effect you describe.
To fix 1, you just need a new washer (two main sizes in UK. One for bath taps, one for basin taps). To fix 2 you will need a new tap insert.... probably have to buy these in pairs complete with the tap tops... typical B & Q offering. Good news is that replacement taptops often come with several different screw insert pieces so you can fit them to your existing tap bodies.
Then it's just a matter of switching off the water supply, and/or draining the hot water tank down depending on whether you have a local isolator near the tap, removing the existing tap top and unscrewing the insert. Check for either 1 or 2 and replace as necessary. Of course, if cutting of the water is a major drama you're more or less forced to assume that you will need a whole insert before you can find out for sure.... unless you happen to live next door to a DIY store.
Good Luck
M