The British- and Irish-English phrase (all) fur coat(s) and no knickers, or no drawers, denotes ostentatious vulgarity in social life, from the literal sense of a fashionably dressed woman whose appearance covers vulgarity.
—Cf. also Queen Anne front and Mary Ann back.

These are, in chronological order, the earliest occurrences of the phrase (all) fur coat(s) and no knickers, or no drawers, that I have found:

1-: From The good neighbours of ‘Corned Beef City’, an article about Dagenham, “this town out on the flatlands of Essex”, by Geoffrey Goodman, published in the Daily Herald (London, England) of Monday 4^th March 1963:

It began 40 years ago with an organised movement of East Enders from London. And its fundamental “one-class” character has hardly changed since then.
[…]
The second and third generations of the original London settlers are moving out to the more “snooty” areas of Ilford, Romford and Woodford—partly because there are no houses for them in Dagenham […].
Those who remain are solidly working class, with their social horizons limited by the Ford skyline and the drabness of the Essex marshland.
The young people who move away […] say:
“We don’t like the place. It’s not that we’re snobs, but we find it a depressing place. After all, it’s a slum clearance estate, isn’t it?”
And the people they join in towns near Dagenham agree with them.
Ilford folk call it “A corned-beef city.”
Dagenham has a gruff, winking, East End Cockney reply: “ln Ilford, they’re all fur coats and no drawers.”

2-: From Edinburgh: Their Noble Town, by Iain Hamilton, published in The Illustrated London News (London, England) of Saturday 27^th August 1966:

The second thing to be said about Edinburgh is that in appearance, temperament, and atmosphere it differs from Glasgow, scarcely fifty miles to the west, at least as much as Bath from Wigan.
The face of Edinburgh is dramatic, and aristocratic; but its aristocracy is of the peculiarly Scottish kind that belongs to ploughman as much as peer. That first impression of theatricality does not diminish with closer acquaintance, but the visitor energetic enough to quarter the city on his feet soon finds that its great stage-sets, unlike those of the theatre or, for that matter, of more pretentious towns where art and money have been lavished upon facades and all behind left shabby, are four-square and full of detail waiting to expand in the imagination and recreate past centuries in the present moment. Whether or not there was any truth in the Glasgow ribaldry directed against the haughtier goodwives of Edinburgh during the depression, “Fur coats and nae drawers!” there is no doubt that their houses are well-found.

3-: From City comment, in The Guardian (Manchester, Lancashire, England) of Friday 15^th September 1972:

Reading the circular put out by stockbrokers Belisha and Co. yesterday on the P and O/Bovis deal, it is hard to resist picturing the two companies as much sought-after ladies in a marriage market.
On the one hand we have the worthy Miss P and O, currently looking a bit threadbare it is true, but full of promise for the future. Belisha […] makes a virtue of this lady’s disadvantages. Yes, of course she is rather inefficient about the house earning a return of only 1 per cent on turnover. But look at the room for improvement.
The contrast with the hussy Mistress Bovis needs no emphasis. There she goes, flaunting her inflated land values and ill-gotten gains in the speculative housing market just at a time when the housing market is heading for a slump. All fur coats and no knickers is the judgment which would be passed on this specimen in more robust climes than the City.

4-: From What really happens when the Pete Murray spectacular hits the road: Uproar in the house, published in the Daily Mirror (London, England) of Wednesday 13^th December 1972:

Five mornings a week more than five million British housewives tune in to Pete Murray’s Radio 2 programme “Open House.” Frequently this request programme moves out of the studio and on to the open road. The Mirror looks at what happened when his show took over Newcastle City Hall.
It’s the “All Feur Ko-ats and Nee Knickers” show up in deepest Geordieland and Newcastle City Hall is alive with the unrestrained laughter of 2,000 happy women.
[…]
Pete had been rehearsing the old Geordie saying: “All Feur Ko-ats and Nee Knickers” half the night and getting his tongue in a twist. It is the ironic comment of one side of Newcastle about the posh side: “All fur coats and no knickers.”

5-: From Reading is ‘worst area for bad debts’, published in the Evening Post (Reading, Berkshire, England) of Friday 4^th January 1974:

The Thames Valley is the worst area for bad debts in the country, the boss of one of Britain’s biggest credit investigation firms has said.
“It’s a floating population and everyone wants to keep up with the Jones’s. All fur coats and no knickers—that’s how I think of the Thames Valley,” said Mr Glyn Lewis, boss of Credit Bureau Ltd.
“House prices have gone sky high and with the high mortgage rates people are living more and more above their means.”
Proposed changes in the rules governing the settlement of debts through the county courts will mean more and more petty defaulters escaping lightly, Mr Lewis feels.
And he confidently expects that a high proportion of the new defaulters will live in the Reading area.

6-: From an interview by Alex Hamilton of the Scottish novelist, short-story writer, poet and essayist Muriel Spark (née Camberg – 1918-2006), published in The Guardian (Manchester, Greater Manchester, England) of Friday 8^th November 1974—the following is about Muriel Spark’s childhood in Bruntsfield, a residential area in Edinburgh:

They lived in an uppish district of Edinburgh, and the phrase about the adjacent Morningside, “fur coats and nae knickers,” could fairly be applied to it.

7-: From The long, hot-air summit, published in the Daily Mirror (London, England) of Monday 9^th May 1977—this newspaper was criticising the lack of results of the summit meeting of the heads of France, West Germany, Italy, Japan, the United Kingdom, the United States and Canada, which was held in London on 7^th and 8^th May 1977:

Jimmy Carter * has mastered one Geordie phrase—“Howay the lads.”
Here’s another for him—“All fur coat and no knickers.”
That was the summit. Glamour galore. But not much underneath.

* Jimmy Carter (James Earl Carter – born 1924), U.S. Democratic statesman, was the 39^th President of the USA from 1977 to 1981.

8-: From the review by Joseph McKiernan of what the British journalist, author and broadcaster John Ardagh (1928-2008) wrote about Newcastle-upon-Tyne in A Tale of Five Cities: Life in Provincial Europe Today (Stuttgart, Bologna, Newcastle-upon-Tyne, Toulouse, Ljubljana) (London: Secker and Warburg, 1979)—review published in the Evening Chronicle (Newcastle-upon-Tyne, Northumberland, England) of Friday 9^th November 1979:

The Tory suburbs of Jesmond and Gosforth may not share Byker’s life style but have the same very English quality of casual togetherness. People are cheerfully vulgar. No one stands on ceremony or tries to impress with rank or title. There is much harmless, bitchy gossiping and promiscuity veiled in hypocrisy.
Jesmond, as seen by the working class, is where the ladies wear fur coats and no knickers.

A variant of the phrase is (all) red hat and no knickers, or no drawers—used for example by the English author Alan Sillitoe (1928-2010) in Out of the Whirlpool (London: Hutchinson, 1987), a novel in which he evokes the slums of the English Midlands. This is from the review of the book, by Andrew Brown, published in the Calgary Herald (Calgary, Alberta, Canada) of Sunday 13^th March 1988:

In the middle of the book Sillitoe’s language abruptly changes from working-class to middle-class. Early on, he uses a wonderful working-class vernacular. Peter’s father, for instance, “swanned off” with a fancy woman “all red hat and no drawers.”