The primary meaning of the British-English phrase to go like a bomb is—of a motorcar, an aircraft, a motorcycle, an animal, a person—to move very fast.
The image is of a speeding explosive projectile, as is clear in the following advertisement for Kellogg’s, published in The Northern Whig and Belfast Post (Belfast, Antrim, Northern Ireland) of Monday 21st November 1955:
Do you start with a bang?
Do you go to work like a zooming rocket, or crawl in like a damp squib? It can make such a difference to your day when you start with a bang, not a whimper!
One way to start right is to eat right at the start. All of us, health authorities now say, need a full one-third of our daily food at breakfast-time. The first meal of the day should never be skimped, never be skipped!
No time? Too much preparation and washing-up for the wife? Get on all right with a quick cuppa and a slice of bread? Can’t face food early on?
All right . . . it’s you saying so. But those who know don’t agree; they say you’re piling-up trouble for yourself.
THE REAL ANSWER
Add a heaping plateful of cr-r-unchy Kellogg’s, with milk and sugar, to your breakfast for a bang-up start to the day! No preparation! No messy washing-up! Eaten in no time, yet digestible as well as sustaining! You need to eat a breakfast you can go to work on.
Do you start with a bang or a whimper? Get on the way up with tasty, crunchy, quick-to-serve, quick-to-eat Kellogg’s—with milk and sugar—every morning! You’ll go like a bomb!
Help yourself to Kellogg’s
SERVED IN A COUPLE OF SHAKES
The earliest occurrences of the phrase to go like a bomb that I have found are as follows, in chronological order:
1-: From the caption to the following photograph, published in The Illustrated Sporting and Dramatic News (London, England) of Friday 25th November 1938—this photograph illustrated the account of the annual Veteran Car Run from London to Brighton:
WHAT IS IT THIS TIME?: Mr. A. Hess, the well-known motoring commentator (at the wheel) and Mr. J. Heitner, Editor of the “Sphere” had some trying times with their 1900 Panhard Sterling Dog-cart, but they got it going “like a bomb” (12 m.p.h. to be precise!) later on and arrived only a few minutes late at Brighton.
2-: From Harry—And “The Flash”, by Anthony Fenwick, a short story published in the Liverpool Echo (Liverpool, Lancashire, England) of Saturday 15th February 1947:
HARRY BATES used to be a porter in Spitalfields Market. It’s not a soft job, and Harry worked as hard as the next. He wasn’t married, and the light of his life was a greyhound called The Flash. All Harry’s spare time was spent out on the Flats, training The Flash to be a great winner.
[…]
[…] She was put in for a race at one of the small London tracks. She went like a bomb from start to finish and came very near beating the track record.
3-: From Cheshire Take-Off Of The First Four-Engine Jet, published in the Liverpool Echo (Liverpool, Lancashire, England) of Monday 6th September 1948:
“GOING LIKE A BOMB”
Pilot’s Message To Chairman
PERFECT TAKE-OFFAviation history was made at Woodford Aerodrome, Cheshire, to-day, when James Orrell, Avro chief test pilot, made a perfect take-off with the new Avro-Tudor 8—the world’s first four-engined jet airliner.
After two days intensive work on the plane Avro and Rolls Royce engineers made a final check-up early this morning and after several run-ups the plane took off on its flight to the Ministry of Supply Aerodrome at Boscombe Downs, Hampshire, where it arrived shortly before mid-day.
Powered by four Rolls Royce Nene jet engines, the plane is to be fitted with special equipment for high altitude research in the substratosphere.
Five minutes after taking off Orrell spoke by radio-telegraph to Sir Roy Dobson, who watched the maiden flight, and said: “She is going like a bomb, handling very sweetly.”
The phrase to go like a bomb and its variant to go down like a bomb later came to mean to be very successful or popular.
Note: This British-English use of the noun bomb contrasts with an American-English use denoting a total failure, from the image of a bomb leaving things in ruins.
The earliest occurrences that I have found of the British-English phrase to go (down) like a bomb—meaning to be very successful or popular—are as follows, in chronological order:
1-: From the column Under the Counter, by Noel Whitcomb, published in the Daily Mirror (London, England) of Wednesday 1st June 1949:
The scene in the foyer of the London Casino where La Day is gambolling in a £50,000 musical called “Latin Quarter,” was pretty gay.
A rumba band jittered as the customers streamed in, and a couple of pretty girls wandered round taking photographs of the playgoers at five bob a time.
This “snapping the customers” is a new idea in London, though I’ve seen it done in Paris.
“We only started tonight,” smiled camera-girl Sonia Heddon, “and the idea is going down like a bomb.”
2-: From The Dam Busters, by the Australian fighter pilot Paul Brickhill (1916-1991)—as published in the Liverpool Echo (Liverpool, Lancashire, England) of Friday 29th February 1952:
One by one they landed and were driven to the ops. room, where Harris, Cochrane and Wallis listened intently. Gibson came in, his hair pressed flat from eight hours under his helmet. “It was a wizard party, sir,” he said. “Went like a bomb, but we couldn’t quieten some of the flak. I’m afraid some of the boys got the hammer. Don’t know how many yet. Hopwood and Maudslay for certain.”
3-: From Come off it, Mrs. Carnegie, by Noel Whitcomb, published in the Daily Mirror (London, England) of Thursday 12th November 1953:
Mrs. Dale Carnegie, wife of the chap who wrote “How to Win Friends and Influence People,” is stealing some of her Old Man’s thunder.
She has written a book telling women how to make their husbands successes. And it’s going down like a bomb with American women.
4-: From Oh, what a night of fireworks!, an account of the Daily Mirror’s Cavalcade of Sport, by Noel Whitcomb, published in the Daily Mirror (London, England) of Thursday 27th May 1954:
The seven-a-side Rugger went down like a bomb—you’d have thought it was an international.
The earliest occurrences of the variant to go down a bomb that I have found are as follows, in chronological order:
1-: From The boys with the beat: Small group—big sound, by Geoff Leack, published in the Liverpool Echo and Evening Express (Liverpool, Lancashire, England) of Saturday 4th July 1964:
The Black Knights are a talented trio who have been on the Mersey beat scene for about eighteen months. A semi-pro group, they manage to play week-ends as far away as Middlesborough [sic], Leeds, Bradford &c. […] The group prefer to play out of town bookings, because apart from the financial advantages, they say that they always “go down a bomb,” and have had requests from fans in six different cities who want to start a fan club for them.
2-: From an article by Patrick Doncaster about the Liverpudlian rock-and-roll singer Tommy Quickly (Thomas Quigley – born 1945), published in the Daily Mirror (London, England) of Saturday 7th November 1964:
I have seen artists far better known than Quickly die the slow and dreadful death on a bill where everyone is fidgeting for the big act to get in front of the footlights.
Last night the big act was the Beatles—and currently there is no bigger act in the world.
Yet Quickly got a tremendous hand. And we have it on the authority of Beatle Paul McCartney that it is much the same every night. “He goes down a bomb,” says Paul.
word histories 
You make the useful point about the difference between British and US English with this phrase. I notice lately, however, that the verb, in BE, is taking on the American meaning, in examples like: “the show bombed” i.e it failed. This is still a bit puzzling to a British English speaker, but becoming more common, I think.
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