Up in Annie's room behind the wallpaper

Every source I could find seemed to place the origin of the phrase as the UK, some time in the early 1900's most likely during World War I

Eric Partridge's Dictionary of Catch Phrases makes mention of "Up in Annie's Room", saying (that form) came to be around WWI, and was originally used as a response to an inquiry of an absent man's whereabouts.

Word-detective goes further, explaining it more specifically as a joking inquiry to the whereabouts of a soldier

from word-detective:

a joking reply to an inquiry (often from a sergeant or other superior) as to the whereabouts of another soldier. As this exchange usually took place either in the barracks or in the field, the humor came from the fact that there was no possible "Annie," let alone a room upstairs in which to dally with her.

According to the latter source the phrase found renewed life among civilians after the war, being used as a joke/absurd location for an everyday object, finding increasingly implausible locations for added humor (thus the behind the wallpaper/clock variant)

Judging by all this it has fallen into apparent disuse, and thus become fairly obscure; hopefully I can find some ham-fisted use for it at the office today...

The earliest Google Books match for the phrase appears to be in Walter Downing, Digger Dialects: A Collection of Slang Phrases Used by the Australian Soldiers on Active Service (1919):

ANNIE (n.)—"Gentle Annie," a big German howitzer, which fired on Bailleul during March and April, 1918. "In Annie's room"—an answer to questions as to the whereabouts of someone who cannot be found (See "Hung on the wire" [the entry for which reads "Absent; missing").

The original phrase thus seems to have been a satirical way of saying "missing for whatever reason." It's hard to gauge the underlying seriousness of the status of having gone missing that the term may originally have been used to suggest, though the phrase certainly became utterly facetious at some point after the war. The alternative "Hung on the wire" is certainly a grim enough alternative.

Michael Quinion, Why Is Q Always Followed by U?: Word-Perfect Answers to the Most-Asked Questions About Language (2009) offers this discussion of "up in Annie's room":

Eric Partridge says that it dates from shortly before [World War I], but was 'at its height during it'. He explains that up in Annie's room was a common dismissive reply to a sergeant or corporal who was asking where somebody was. The implication was that the person sought wasn't just elsewhere but actively didn't want to be found. ...

It was after the war ended when the phrase had been taken back into civvy street that behind the clock was added. This makes more sense than you might think — it was common practice in homes to put bills or letters behind the mantelpiece clock as an informal filing system so they could be found when needed. Another, more fancifully extended, form is up in Annie's room, behind the wallpaper. The expression was taken [from British Army slang] to Australia — its first appearance in print was in W. H. Downing's Digger Dialects of 1919. A later Australian elaboration is up in Annie's room resting on a pedestal. Dart players borrowed up in Annie's room for the double-one.

It's not altogether clear which of Partridge's books Quinion is citing here. The first edition of A Dictionary of Slang and Unconventional English (1937) has this:

Annie's room (up) in. A military c[atch]p[hrase] reply to a query concerning someone's whereabouts : military, slightly pre-G[reat] W[ar]. The original implication being that he was 'a bit of a lad'. Cf. hanging on the barbed wire [defined as "A military c[atch]p[hrase] reply to an inquiry as to a man's whereabouts : 1916–18. ... Ex men left dead on the wire after an attack."]

Terence Dolan, A Dictionary of Hiberno-English: The Irish Use of English, revised edition (2004) claims "up in Nelly's room behind the wallpaper" as an Irish variant:

Nelly n., a woman's name (of a type that may suggest a maid-servant), in the expression 'up in Nelly's room behind the wallpaper' (or 'behind the wardrobe,' etc.), a sarcastic reply to an exasperating question. 'Where's my cap?' — 'Up in Nelly's room behind the wallpaper.'