This "blunder" meaning of cock-up has been used before the 1960s, from at least the 1940s in writing.
It can be found in the 1950 Sea slang of the twentieth century: Royal Navy, Merchant Navy, yachtsmen, fishermen, bargemen, canalmen, miscellaneous by Wilfred Granville, which covers the period from 1900 to 1949.
cock-up. A mess-up, a bungled piece of work. A LASH-UP. (Lower-deck.)
lash-up. General confusion caused by a misunderstood order or a bungled job of work. Cf. the Royal Navy's lower-deck term, COCK-UP.
Pierre Clostermann's 1948 Le Grand Cirque is one of the very first post-WWII fighter pilot memoirs:
Hullo Filmstar Leader, sorry old boy, there is a cock-up about the Typhies. Do your best if you can without !
Workers in Stalin's Russia by M. L. Berneri (1944):
The journey of approx. 500 miles took us five days, and has been known to take ten days. As we had to take food for this time we travelled rather like a person moving house. There was a cock-up about transport to take us across the ice to the station
Finally, the term isn't particularly offensive. It's been used scores of times in UK parliament, most recently by Peter Bone:
Last Sunday I attended Indian republic day at the Wellingborough Hindu Association, yet the same week we learn that a £20 billion fighter contract has been lost to, of all people, the French. We now know that the lead bidder was not the British Prime Minister or the British Government, but the Germans. What on earth do they know about cricket and curries? Why was the British Government not leading on that? How did the Secretary of State allow such a cock-up?
The Facts on File Dictionary of Cliché by Christine Ammer says:
not for all the tea in China Not at any price. Eminent lexicographers [such as Eric Partridge] agree that this term originated in Australia in the 1890s and soon spread to the rest of the tea-drinking English-speaking world. The OED cites K. Tennant's Ride on, Stranger (1943): "I'm not going to stand in my girl's light for all the tea in China."
The earliest result for "all the tea in China" in Google Books is in 1895's The Amateur Fisherman's Guide by Charles Thackeray. There's no preview so it cannot be verified, but it's plausible as Thackeray also founded the Amateur Fishermen's Association of New South Wales in 1895.
The earliest result The Phrase Finder can find to verify the date is from an Australian 1914, but I found an earlier instance in New Zealand's Marlborough Express of 9 September 1907 (actually in a "What The Papers Say" section, credited to the Post):
A short time ago the Celestial maiden would not exchange one of her tiny, disfigured feet for all the tea in China; if she lives long enough she will probably weep salt tears of regret that she is not in the fashion, and, even though respect for one's elders is commoner in China than it is in, say, New Zealand, may in her heart feel bitterness against her parents for not having been up-to-date.
Best Answer
Eric Partridge, A Dictionary of Slang and Unconventional English, second edition (1938) has the following entry for the phrase:
Partridge also has entries for "go and boil your head" and "go and eat coke," so the "go and" part of the phrase seems to have been a recurring formula for introducing a dismissive recommendation.
The phrase "take a running jump" simply refers to attempting an informal long jump (as opposed to a standing broad jump, which involves leaping from a standstill). However, an interesting example from Canada suggests that "taking a running jump" might be used to imply "running headlong [into something] without exercising appropriate caution" or, more simply, "acting precipitously." From Official Report of Debates, House of Commons [of Canada] (1932) [combined snippets]:
A Google Books search turns up several relevant matches for "go and take a running jump" from the 1950s. From an unidentified article in Medical Press (1950) [combined snippets]:
A Mr. Dillon of the Irish Parliament uses the expression on multiple occasions, in 1950 the in 1952. From Parliamentary Debates [of Ireland]; Official Report (1950) [combined snippets]:
And from Parliamentary Debates [of Ireland]; Official Report (1952) [combined snippets]:
An earlier, related instance appears in an unidentified article in Blackwood's Magazine (1936) [combined snippets]:
A search of the British Newspaper Archive yields two matches for "go take a running jump at himself" from 1931 and 1932, both of which appeared in the same London newspaper. From "Mr. (or Miss) Wamble," in the [London] Bystander (June 17, 1931):
And from "Cedric Belfrage discusses the Passing Picture Shows," in the [London] Bystander (September 7, 1932):
Conclusion
The earliest print instances I found of the invitation to "go and take a running jump at [oneself]" are from 1931 and 1932 in a London newspaper. Partridge identifies the expression as a catchphrase less than a decade later, and I see no reason to doubt him on this point.
One feature of catchphrases (most memorably itemized by Charles Mackay in Memoirs of Extraordinary Popular Delusions and the Madness of Crowds [1852]) is that they need not possess any historical point or clarity of meaning in order to capture the popular imagination—hence, Mackay's observations about the successive vogues of "Quoz!" and "There he goes with his eye out!" and "Has your mother sold her mangle?" during the first half of the nineteenth century.
Perhaps the phrase "go and take a running jump at yourself" originated as an extemporaneous expression of scorn that caught on specifically because it doesn't really make sense.