The answer that occurred to me was “wash up” (as opposed to just “wash”). I think of that as a refined British expression for bathing or showering. But research did not exactly put that beyond all doubt.
You’d start work about six. We usually got out around maybe dark or seven or eight, nine o’clock. I come back as late as ten o’clock at night. Sometimes I just laid down to sleep, not even sleep – then wash up.
-Working, by Studs Terkel
“Pa,” she called. “John, git up! You, Al. Git up an’ git washed.” Startled sleepy eyes looked out at her. “All of you,” Ma cried. “You git up an’ git your face washed. An’ comb your hair.”
Uncle John looked pale and sick. There was a red bruised place on his chin.
Pa demanded, “What’s the matter?”
“The Committee,” Ma cried. “They’s a committee – a ladies’ committee a-comin’ to visit. Git up now an’ git washed. An’ while we was a-sleepin’ an’ a-snorin’, Tom’s went out an’ got work. Git up, now.”
They came sleepily out of the tent. Uncle John staggered a little, and his face was pained.
“Git over to that house and wash up,” Ma ordered.
-The Grapes of Wrath, by John Steinbeck
Both of those quotations seem to show people using the expression “wash up” for washing themselves, as opposed to washing dishes, for example. But the people using the expression seem to be rural Americans who are not particularly fastidious about language. And I did find lots of uses of “wash,” with our without “up,” for cleaning house, and none of them British. So all I can say is that to my own ears, “washing up” (with no stated object) sounds like a slightly stilted expression for bathing or showering.
P.S. I cannot find the script online, but I believe that in the 1984 film The Razor's Edge, Elliott Templeton asks Larry Darrell to "have a wash" before attending his party.