I am looking for the full lyrics of a song I heard on the radio
several times in London, UK, in about 1961.  The first line was "Upon
the monday morning, oh".  It was sung by a man.  I don't remember if
it was accompanied by any instrumentals, but if it was it must have
been minimal as I remember only the song.

I am confident that your song is "All the While," which begins with this verse:

Upon a Monday morning-O, the rain it was a-raining;
My love she came to me and said,
  Oh, when will you and I be wed,
  For I have bought a double bed
  And Mother is complaining-O.
And all the while the rain it was a raining.

The complete lyrics may be found here:

Digital Tradition: All the While
http://sniff.numachi.com/~rickheit/dtrad/pages/tiALLWHILE.html

"All the While" was written by Myles Rudge and Ted Dicks as a novelty
song for the British performer Miles Cribbins. Cribbins recorded it
under the title "Folk Song." It appears on Cribbins' albums "Carry On
Cutler" and "The Very Best Of Bernard Cribbins."

Here you may purchase "The Very Best of Bernard Cribbins":

Streetsonline
http://www.streetsonline.co.uk/common/product.jhtml?pid=50209475

Amazon
http://www.amazon.com/gp/product/offer-listing/B0001ZXM3G//103-8705472-4408639?condition=all

Another recorded version was by the Modern Folk Quartet. The song
(titled "And All the While) appears on the MFQ album "Changes." This
was released on vinyl in 1964 (Warner Bros. Record WS-1546). A
Japanese CD of the album was released in 1990, but it is very rare and
difficult to find (Japan, Warner Bros. Record WPCP-3423).

You can purchase a used vinyl LP of the MFQ album "Changes" here:

Musicstack
http://www.musicstack.com/item.cgi?item=369037

If this is not the correct song, or if anything is unclear, please
request clarification; I'll be glad to offer further assistance before
you rate my answer.

Best regards,
pinkfreud

The answer was exactly correct and I really appreciated the additional
information provided on where to get recordings etc.

Please forgive an error in my text above. I referred to Bernard
Cribbins as "the British performer Miles Cribbins." Bernard is the
correct name. My mind was still on the songwriter Myles Rudge, I
suppose. Too many Miles. ;-)

Thanks for the five-star rating and the tip!

~pinkfreud