Quoting from www.cutterandtailor.com/forum/index.php?showtopic=381&hl=umsie:
"An "umsie" or "ums" is a name given in old tailoring firms to a new apprentice. The reason is that you cannot remember the new person's name, "um, what's the name again?" - ergo, "Umsie". The young apprentice would sometimes be addressed as that to their face."
Possible origin: In English we might say, "Um, you there, do this...."; likewise, in German a sentence could begin, "Ummm, Sie..."
Also, from the "Savile Row Jargon" section in Richard Walker's The Savile Row Story, Prion, London, 1988:
"Umsies: Someone being talked about who (sic) the speaker does not want to identify directly, because he is present."
The word is reported also to be used in a naval context, but I haven't found a reference.
"This old hierarchical business of starting as the "umsie" sweeping the floors and sewing buttons before spending 7 years learning to make pockets, should be updated. " (from: www.askandyaboutclothes.com/forum/showthread.php?88791-Best-steps-to-becoming-a-tailor)