(Thanks to Nancy Friedman.) Evasive, tricky, artful; dubious, unreliable. OED’s first citation is 1861. Google Ngrams show British use taking off in about 1940 and American, characteristically for NOOBs, circa 1990.
To heighten the fun of the chase, she gives Grace a road buddy in Darcy Kohler, a dodgy market analyst who stands to lose her condo if her missing boyfriend can’t be found to make good on the bond she co-signed. (Marilyn Stasio, New York Times, September 18, 1988)/ I see that the administration is pressing New York’s attorney general to drop its investigation into dodgy foreclosure practices and settle with the banks. (Megan McArdle, TheAtlantic.com, August 25, 2011)
notoneoffbritishisms.com 
In everyday speech, something that is dodgy is likely to be bordering on illegal, or highly likely to be a scam of some kind e.g. a dodgy business deal. It’s often colloquially referred to as “a bit dodge” and usually indicates that you shouldn’t put your trust in the deal, or the person making it. It can also mean that something is broken or not quite right, i.e. the chair was dodgy because the leg was a different length to the others.
I wondered if “dodgy” is beating back “bogus.” A quick check though (http://goo.gl/Dy9uD) shows this is not the case. Instead, we see a secular increase in dodginess/bogosity. I attribute this to the second law of thermodynamics.
We don’t need “dodgy” when we’ve got a good old Americanism like “hinky.”
Sheesh.
“Hinky” doesn’t sound quite the same to me, but in my observation “sketchy” is an extremely close US equivalent in both connotation and use. (Although it would be, “totally sketch” rather than “a bit dodge”.)
Dodgy – can be used to describe something of questionable character, manner or capability.
Potential uses – ‘I don’t like you hanging around with Steve, he’s a dodgy character, upto no good drink & drug driving and his fixation with guns is dodgy too’ implicating a ‘dodgy person.’
Dodgy manner – ‘I like all your friends except Sue. She seems a little ‘dodgy.’ She doesn’t say much and creeps around’ ‘She’s ok, just a little quiet and shy.’
A potential use for ‘dodgy’ referencing capability – ‘I don’t think I can train tonight, my knee still feels a little dodgy’ or ‘Don’t use the car, it’s going into the garage, the brakes are dodgy as they slipped in the rain last night. It’s getting picked up later.’
“Dodgy” is not the same as “bogus”. A dodgy policeman is making a little bit of money on the side. A bogus policeman would be someone pretending to be police who wasn’t.
Q. “How do I get to Leicester Square?”
A. “Oh, it’s a bit dodgy.”
I’ve done absolutely no reaserch into this, but I was thinking of “the srtful Dodger” from Oliver Twist…published in the 1830s, so well before the OED’s first citation. Dodger…artful, someone who is clever, or crafty…Dodger, some one who escapes or evades…I wonder if Dodger gave birth to the expression? A dodgy person?
Yet another variation on use of this word in the UK is in reference to food which has caused stomach upset or other illness. We would often refer to a dodgy curry/burger/kebab, implying that it is to blame for a bout of D & V.
‘Dodgy’ was a catchword popularised by a fellow called Norman Vaughan who presented the ITV show ‘Sunday Night at the London Palladium’ in the 1960’s.
One of the canonical (UK) usages spotted in this piece.
https://krebsonsecurity.com/2022/09/say-hello-to-crazy-thin-deep-insert-atm-skimmers/
“So keep your wits about you when you’re at the ATM, and avoid dodgy-looking and standalone cash machines in low-lit areas, if possible.”