“You are never more than a few feet away from a rat” is said about several big cities, but most specifically about London. “You know that wherever you are in London, you’re only 30 feet away from a rat” was said in the film Naked (1993), directed by Mike Leigh.
 
“Rachel had heard that in NYC or any major town you’re never more than ten feet away from a rat” was cited in print in 1997.
 
   
Naked (1993)—film script by Mike Leigh
You know that wherever you are in London, you’re only 30 feet away from a rat.
 
Google Books
The Keys to the Street
By Ruth Rendell
New York, NY: Random House
1996
Pg. 127:
Another thing he’d heard was that no matter where you were, city or countryside, you were never more than six feet from a rat.
 
18 October 1997, The Orange County Register (Santa Ana, CA), “British rats are going from sewers into homes” by Sarah Boseley (The Guardian), pg. 25, col. 3:
“You are never more than five meters (16 1/2 feet) from a rat, it is said. True or not, in urban areas there are plenty of rats around, while in the countryside you wouldn’t be able to find a piece of hedgerow without a rat running along it in any 24-hour period,” he said.
 
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Rachel’s Machine
By Martin Wagner
London: Pinter & Martin
1997
Pg. 140:
Rachel had heard that in NYC or any major town you’re never more than ten feet away from a rat.
 
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Legacy
By Alan Judd
New York, NY: Alfred A. Knopf
2001
Pg. 70:
Charles had read that you were never more than about three feet from a rat in London. Maybe it was six feet, or sixty.
 
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Chimps
By Simon Block
London: Nick Hern
2005
Pg. 28:
Wherever you are in London it’s said you’re never more than seven feet from a rat.
 
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Whittington
By Alan W. Armstrong
New York, NY: Random House
2005
Pp. 82-83:
“In London you’re never more than six feet from a rat.”
 
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Nothing Like the Night
By David Lawrence
New York, NY: Thomas Dunne Books (St. Martin’s Press)
2005
Pg. 21:
In London, you’re never more than twenty feet from a rat.
 
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The Traveler’s Handbook:
The Insider’s Guide to World Travel
Edited by Jonathan Lorie and Amy Sohanpaul
Guilford, CT: Globe Pequot Press
2006, ©2005
Pg. 137:
They say that in London, you’re never more than five feet from a rat.
 
Google Books
Banksy Locations & Tours:
A Collection of Graffiti Locations and Photographs in London, England.
Edited by Martin Bull
Oakland, CA: PM Press
2011
Pg. F1:
The photo of it on Banksy’s website calls it a “ratapult” and it was accompanied with the following text: “RAT FACT — In London you’re never more than 20 feet away from somebody telling you you’re never more than 20 feet away from a rat.”
 
The Daily Journalist
December 29, 2012
164ft Zombie Rat Myth Bust
By Mike Sutton.
Mythbusting the mythbusters and their super myth making
On the 17th December the BBC (Prichard, C. 2012 ) set about debunking the old maxim that “you are never more than six feet away from a rat”.
 
The Six Feet From a Rat Myth was formulated in 1909 on the basis of the unfounded belief that rural Britain had one rat per acre. Since there were 40,0000 acres of cultivated land and a population of 40 million people, the Six Foot From a Rat Myth was born.
     
NewStatesman
London has more branches of Starbucks than any EU country
And not one of them is in Hackney.
BY JONN ELLEDGE PUBLISHED 28 MAY, 2014 - 18:39
(...)
In London, the old adage goes, you are never more than 20 feet from a rat, and never more than 50 from a decaf soy latte.