Welch is an archaic form of Welsh, one that pretty much only survives in the sense of to fail to pay out on a lost wager. This usage dates to around 1857 when it appears in the 8 June issue of the Morning Chronicle:
He got his living by “welching” and taking in the “flats.”
The origin is obscure, but it undoubtedly originated as an ethnic slur of the Welsh people.
(Source: Oxford English Dictionary, 2nd Edition)
Copyright 1997-2013, by David Wilton