The word is “welched”, a betting term, meaning to default on payment upon losing a bet.
Where did Macolme get his info?
Cannot figure out which mailbag article is being referenced, though I have a vague recollection of one.
As to the OP, “welch” is a variant of “welsh.” And as a verb it is directly related to stereotyping about the Welsh. See most any dictionary. 
Here it is: Why can’t Prince Rainier become a king? (25-Dec-1992), and it is indeed a column by Cecil.
In the column, Cecil says, in apropos of Monaco’s independence, “during the French Revolution France welshed on the deal.”
A comment on the column was posted later:
As DSYoungEsq says, a dictionary can usually clear up confusion. From Merriam-Webster:
Also from Merriam-Webser:
Nothing in those references mentions a connection to Wales.
“Etymology: probably from Welsh, adjective” - the capital letter indicates the proper noun.
I have been searching online for a direct link between welshing on a bet, and Wales. So far, no joy. My Concise O.E.D. says “origin unknown.”
As someone who is Welsh I did some research of the term, and I believe the terms welch/welsh are connected as an ethnic slur. I thought they just might have been similar words that become connected, but it appears not. Welch is just a corrupted form of Welsh.
According to this it originated in betting where English bookmakers to avoid paying large debts fled over the border into Wales and were said to have gone Welsh.
So the slur is directed at the English then. It does refer to Wales, but only as a name of a country, not as a comment on the character of it’s inhabitants. Right?
Man, would you folks all settle down? Here, have some toast with cheese sauce on it. Tastes like rabbit!
Wales as the home of thieves? Someone who reneges on a bet has “gone Welsh”, which implies all Welshmen are thieves or cannot be trusted at least.
How dare you! This is a foul calumny. We only use real cheese.
The English bookies were the thieves. Wales was their destination, simply as the next country over. That’s how I read your cite, anyway.
Since my O.E.D says the origin is unknown, I’ll take it over a message board post. 
Speaking of message board posts, here’s one with a completely different explanation:
:dubious:
I thought “welch” meant to smear with grape jelly?
:rolleyes:
“Taffy was a Welshman
Taffy was a thief.”
Shakespeare’s notion of a Welshman seems to have been different; he seems them as either pedants or phony psychics.
…or to be a so-called “Christian” who wants to improve on Jesus’s morals?
It’s from the German verb “welken” which means “to fade”.*
Refers to the habit of bookmakers to “fade away” when it’s time to pay out.
Sod all to do with the taffs, even though it’s broadly accurate. 
*well, ok… maybe it is, about as valid as any other theory though!
So after all that, we’re not sure whether it’s an ethnic slur or not? What a gyp.
We know that it is a version of the word “Welsh.” We know that it relates to a practice that is not favorable. We appear unaware for certain how that use of “welsh” came about, though one can draw certain conclusions about probability. With much of the origin of phrases or words, that’s about all one can manage, as samclem is often at pains to point out.
As opposed to…?