Changing a tap washer costs £60 to £95 including VAT in London and £50 to £70 outside it. Those figures sit on the floor of what any attended plumbing visit costs. The reasons the floor is where it is are set out in the answer on why plumbers are so expensive in the UK, and they come down to an hour of a qualified person's time rather than the couple of pounds the washer costs.
This page assumes you have already worked out that a washer is the fault. If you have not, the answer on fixing a dripping tap covers how to tell a washer from a cartridge or a valve, because a quarter-turn or single-lever mixer has no washer in it to change.
Doing it yourself
A washer on a traditional tap is within reach of most households, and it is one of the few plumbing jobs where that is true.
The sequence is short. Isolate the supply. Open the tap fully to drain the standing water and relieve the pressure. Put the plug in, so that nothing small goes down the waste. Lift or unscrew the head cover, which on many taps is a decorative cap concealing a single screw, and take the headgear out with a spanner, holding the tap body so you are not twisting it against the basin. The washer sits at the bottom of the headgear, usually held by a small nut. Swap it, reassemble in reverse, and open the stopcock slowly rather than all at once.
Two things are worth buying before you start: an assortment of washer sizes, because you will not know which you need until the tap is apart, and a replacement O-ring for the same spindle while you are in there.
The part that goes wrong is the isolation
The failure mode on this job is almost never the washer. It is turning the water off.
If there are isolation valves on the tails under the basin, this is a low-risk job, because a screwdriver turns them and only that tap is affected. If there are not, the mains has to come off at the stopcock, and Which? warns that older brass stopcocks seize with limescale and that a seized spindle snaps cleanly when forced. Snapping a stopcock while attempting a trivial repair leaves you with the water off and an emergency call-out, which is comfortably the most expensive outcome available.
So test the stopcock gently, before you buy any parts. If it will not turn, that is the job to book, and while a plumber is fitting isolation valves under the basin, the washer takes them another few minutes.
When a new washer does not stop the drip
This is the common disappointment, and it is worth knowing in advance rather than at the second attempt.
The washer closes against a brass seat inside the tap body. Over years of limescale and grit that seat corrodes or scores, and a new washer cannot seal against a damaged surface. The drip resumes within days, or never stops at all.
The fix is re-seating: cutting the damaged surface flat again with a seating tool, which is a plumber's job rather than a household one, or fitting a nylon seat-and-washer set that sits over the old surface. On old taps neither is always worth doing. Checkatrade puts replacing a tap at £140 average labour, in a range of £60 to £180, on a basis the guide does not state as either including or excluding VAT. Where the tap is corroded, stiff and dated, that is often the sounder purchase.
Ask the plumber to check the seat before refitting. It costs nothing at the time and it is the difference between one visit and two.
Sizing the quote
For a single washer, expect the London bracket above and no more. It is a first-hour job by definition.
If a figure comes back well over £95 with nothing else on the ticket, ask what it covers. There are three usual answers, and all are legitimate: a call-out charged on top of the labour rather than inside it, an out-of-hours or same-day slot, or a plumber pricing for the cartridge they suspect they will find when the tap comes apart. Ask which, and ask what the figure becomes if the job turns out to be the simple version.
Then use the visit. The travel, the parking and the call-out are paid for whether the plumber is at your sink for twenty minutes or ninety, so a stiff second tap, a slow-filling cistern or a missing isolation valve costs little more than the labour it takes. Bundling is the only real saving available on work this small.
Where the part is trivial the choice of person is the whole decision.