The colloquial British-English phrase to sell the family silver, and its variants, mean:
– to sell, or part with, a valuable resource or asset for immediate advantage;
– in particular: to dispose of a nation’s assets for financial gain, for example by privatising a state-run industry.

The noun family silver is used figuratively to designate something considered to be of great value, materially or otherwise—for example in the following from Avery’s Show On Architecture Wide Ranging, by Paul Goldberger, published in The New York Times (New York City, New York, USA) of Saturday 27^th November 1976:

Columbia’s Avery Architectural Library is not merely one of that university’s great treasures, it is also one of the city’s. Under the guidance of the gracious and erudite Adolf K. Placzek, Avery has become not just the finest architectural library in the country, but, in the opinion of many scholars, even the finest in the world.
[…] With no particular purpose except to give the public a bit of pleasure, 45 of the finest drawings in the collection have gone on display in the rotunda of Low Memorial Library at Columbia […].
The exhibition, which Mr. Placzek calls “showing off the family silver,” deliberately eschews any polemical purpose.

The earliest occurrences that I have found of the phrase to sell the family silver and variants are as follows, in chronological order:

1-: From the column Notebook, by Hamish McRae, published in The Guardian (London and Manchester, England) of Thursday 3^rd May 1979—in the title, “stately home” refers to the P&O building, and “family silver” to P&O’s stake in the Beatrice Field:

By all means sell the stately home, but please don’t flog the family silver

P&O’s strategy appears to be to run away from energy as quickly as possible and concentrate its management time on the traditional shipping and property. The P&O building itself is part of this plan: the company will be moving out of the prestigious three floors in the building which carries its name into more modest quarters.
As already announced Lord Inchcape ¹ made it clear that P&O intends pressing ahead with the sale of its 15 per cent stake in the Beatrice Field—possibly to BNOC—as well as its North American gas and oil licences. […]
[…]
Looking through P&O’s present diverse pile of activities the 15 per cent stake in the Beatrice Field looks like the jewel in its crown. Some might suggest it is a brighter prospect than the investment properties which Lord Inchcape is so enthusiastic about.

¹ This refers to Kenneth James William Mackay (1917-1994), 3^rd Earl of Inchcape, Director (1957-1983), Chairman (1973-1983) and Chief Executive (1978-1981) of P&O.

2-: From Big firm welcomes Old Mother Hubbard depleting cupboard, published in the Evening Sentinel (Stoke-on-Trent, Staffordshire, England) of Friday 8^th June 1979:

It came as a surprise to see the enthusiasm with which the chairman of our computer giant I.C.L. is looking forward to the day when the National Enterprise Board unloads its stake in I.C.L.
[…]
The willingness of the Labour Government to sell 17 per cent. of B.P. in the summer of 1977 for £540 million demonstrates that there is no great principle at stake in selling off these interests.
But neither is it justifiable for the Conservatives to make out that in some way they are financing the Budget deficit by selling off assets.
That’s like selling the family silver to pay for the groceries.

3-: From Follow the Road line, by Ivan Fallon, City editor, published in the Sunday Telegraph (London, England) of Sunday 21^st June 1981—context: the British Government wanted to sell the State-owned National Freight Company:

The package he ² has put together is attractive to everyone: the Government gets its £50 million, and achieves its objective of removing one more State company from around its neck. Norman Fowler ³ escapes even the criticism of selling the family silver to the greedy financiers of the City: only about 20 per cent of the £6-£7 million of equity money will come from the Square Mile; the rest will be found from the management, lorry drivers and even secretaries of the company. Even the unions are pleased.

² “He” refers to Peter Thompson, chief executive of the National Freight Company.
³ This refers to Norman Fowler (born 1938), Secretary of State for Transport from May 1979 to September 1981.

4-: From A new Howell down on the farm, by Ivan Fallon, City editor, published in the Sunday Telegraph (London, England) of Sunday 28^th June 1981:

One hears less and less the argument that selling off the big State energy interests is equivalent to selling the family silver.

5-: From the Cambridge Evening News (Cambridge, Cambridgeshire, England) of Wednesday 21^st October 1981:

Surprise sale

We must admit we are still wondering why the surprise announcement to sell off most of the country’s stake in North Sea Oil has been greated [sic] with such general enthusiasm. But we also note that there has to be a great deal of emphasis on the need to convince people who would otherwise “misunderstand.”
One newspaper this morning goes so far as to liken the sale of this asset not to the “flogging off” of the family silver, but rather its “sharing out among the family,” where it will be “kept in better shape.” […]
[…]
[…] We think most people will see this not as selling off the family silver, which has some practical but more aesthetic value; they will believe instead that the Government is selling off the family cow—nicely fattened.

6-: From Bill ‘increases risk of gas explosions’, by David Gow, Labour correspondent, published in The Scotsman (Edinburgh, Midlothian, Scotland) of Wednesday 31^st March 1982—Merlyn Rees, Shadow Energy Secretary, was reported as declaring the following about the Oil and Gas (Enterprise) Bill privatising the gas supply industry, which the Government was rushing through Parliament:

Mr Rees said that if there had been a “rip-off” over Amersham, the “rip-off” enshrined in the Bill would be the “sale of the century.” It was like selling the family silver to pay the butler’s wages.

7-: From Selling a part of the action, published in The Guardian (London and Manchester, England) of Tuesday 26^th July 1983:

They do not get much crisper than the answer Mr Nigel Lawson ⁴ gave the House yesterday on the Treasury’s plans for an additional £500 million of asset sales this year. […] Such asset sales make little sense either in terms of this Government’s—any government’s—objectives for improving the efficiency of British industry or in terms of the state’s own finances.
[…]
[…] Mr Lawson has been duly criticised for “selling the family silver to pay the grocery bill”.

⁴ This refers to Nigel Lawson (1932-2023), Chancellor of the Exchequer from June 1983 to October 1989.

8-: From the transcript of a debate on wealth creation and taxation, which took place in the House of Lords on Wednesday 23^rd November 1983—from Hansard, the official report of all Parliamentary debates:

Lord Boyd-Carpenter: A criticism has been made which I ought to answer. It was made in another place by a member of the party opposite, who said that to sell these nationalised industries and use the proceeds for current expenditure was like “selling the family silver to pay the food bills”. There is a slightly feudal ring about that observation. But with respect, I think it is right. It is extremely important that we should not use the proceeds of these sales simply to finance current expenditure. The resources should be made available for investment in the private sector, either indirectly by using them for necessary Government capital expenditure, so diminishing the demands of the Government on the market for funds, or perhaps even more firmly and effectively by reducing taxation, but certainly not by financing current expenditure.

9-: From Industrial policy ‘key to success’, published in the Liverpool Echo (Liverpool, Merseyside, England) of Wednesday 21^st December 1983:

The Government had “an historic opportunity” to reform key areas of the British economy by returning them to private hands, Mr. John Moore, Financial Secretary to the Treasury, said to-day.
Mr. Moore claimed in a pamphlet Why Privatise? published by the Conservative Political Centre, that the Government’s motives for privatisation were frequently misinterpreted.
“We are accused of sacrificing the nation’s assets for short-term gain and selling the family silver to pay current debts. Nothing could be further from the truth.”

10-: From the transcript of a debate on the Telecommunications Bill, which took place in the House of Lords on Thursday 9^th February 1984—from Hansard, the official report of all Parliamentary debates:

Lord Bruce of Donington: Even though the Bill passes, there may be disenchantment with it and the issue may fail, thus defeating the whole purpose of the Chancellor of the Exchequer in realising capital assets—selling the family silver to pay for the weekly food, which is not reckoned to be the best housekeeping. Sooner or later the public will realise this.

11-: From the transcript of a debate on the Telecommunications Bill, which took place in the House of Lords on Tuesday 21^st February 1984—from Hansard, the official report of all Parliamentary debates:

Lord Bruce of Donington: We all know perfectly well—and it has attracted public comment far outside the cloisters of Government—that it is not good housekeeping to sell the family silver in order to pay the wages of the butler. This is one of those elementary Victorian values to which the Prime Minister so often and so eloquently refers. It is not good policy to sell off assets in order to pay for revenue expenditure; and, sooner or later, this will catch up with the noble Lord. […]
[…]
Lord Cockfield: The noble Lord’s argument about the family silver is based upon a total fallacy. What is happening is that the ownership of British Telecom is being transferred to the British people. There is no reduction in wealth. It is much better for the British people themselves to own an industry of this kind than for it to be owned by one of these amorphous state corporations.