A cliché (probably never actually said) in which characters in the farce are attempting to cover up an embarrassing incident while an important/respectable personage is in the room.
Unfortunately “Whitehall farce” has become such a hackneyed political columnists’ phrase that I can’t actually find anything about the actual plays, and there’s no Wikipedia entry. You might look up Alan Ayckbourn, but I think he followed Whitehall farce as an homage, rather than contributing directly to the genre, which I dimly recall from my drama degree as being huge in the 1950s.
I don’t know if this has anything whatsoever to do with it, but the juxtaposition of “tea,” and “vicar,” and wily shenanigans in an indisputably English setting arises in Gilbert & Sullivan’s 1877 operetta, The Sorcerer.
At a village feast celebrating the betrothal of his son, Sir Marmaduke urges his guests to enjoy themselves and to partake in a spot of tea:
The chorus chimes in:
There is, to the best of my knowledge, no implication being made here that the Vicar has spiked the tea with alcohol. That the tea actually has been spiked (with a powerful love philter, as it happens) is a plot point as yet unknown to Sir Marmaduke and the members of the chorus.
Whether the eschewal of alcohol in favor of tea on the part of the clergy was a common image in the Victorian mind, I cannot say with any certainty. But that would be my guess. Mr. Gilbert, possessed of a keen eye for satire, seems to be banking on the recognition of this characteristic as the basis of a joke that otherwise makes no sense.