The British-English noun Taffia, also Tafia, is applied jocularly to any supposed network of prominent or influential Welsh people, especially one which is strongly nationalistic.

Probably coined on various occasions by different persons, independently from each other, this noun is a blend of:
– the noun Taffy ¹, denoting a Welshman;
– the noun Mafia, denoting any group regarded as exerting a secret and often sinister influence.

¹ Taffy represents a supposed Welsh pronunciation of the given name Davy or David (Welsh Dafydd).

The earliest occurrences of the noun Taffia, also Tafia, that I have found are as follows, in chronological order:

1-: From Council upset by national newspaper story, the account of a meeting of the planning and development committee of the Borough of Afan Council, published in the Port Talbot Guardian (Port Talbot, Glamorgan, Wales) of Friday 19^th July 1974:

Coun. Graham Jones said, “[…] there seems to be a vendetta against the Port Talbot area by some of these newspapers and the T.V. people. The other day a national newspaper ² carried an article about the ‘Welsh Taffia’ and so on.”

² I have not found which national newspaper published the article in question.

2-: From Britons thrive on Brussels Good Life, by Ronald Payne, published in the Sunday Telegraph (London, England) of Sunday 11^th May 1975—the following is about the Commission of the European Community in Brussels:

There are a number of Welshmen in important posts at the commission, including Gwyn Morgan, chef de cabinet to Commissioner George Thomson, and because they are so active they are known waggishly in the buildings as the “Taffia.”

3-: From How Town Hall Taffia was defeated, by George Brock, published in The Observer (London, England) of Sunday 24^th October 1976:

[Emrys] Roberts ³ was the leading figure in Plaid Cymru’s minor revolution this year. In May’s council elections his party took 21 of Merthyr Tydfil’s 33 seats and abruptly ended 70 years of rock-solid Labour control.
[…]
A Plaid Cymru councillor said: ‘I expect you’ve heard they called it the “Taffia.” They joked that Al Capone ⁴ wouldn’t have got a job as a tea-boy in the Town Hall. It was only just an exaggeration.”

³ The Welsh politician Emrys Roberts (born 1931) is a member of Plaid Cymru, the Welsh Nationalist party.
⁴ Alphonse Capone (1899-1947) was a U.S. gangster of Italian descent.

4-: From Not the Godfather, by Clem Thomas, published in The Observer (London, England) of Sunday 27^th March 1977:

If in a rugby sense it is good to be Welsh at the moment, it is also a time charged with a ferocious responsibility. It was inevitable that the fortunes of the international championship, in which Wales narrowly beat the other three home countries, would be intertwined with selection of the 1977 British Lions for New Zealand. But if the Welsh were entitled to a lion’s share, not even in their wildest optimism could they have envisaged such an embarrassment of riches—coach, captain and 16 Welsh players, which perhaps is too heavy a burden. […]
[…]
The captaincy is in good hands, but Bennett ⁵ must guard against the possibility of a Welsh Taffia, and I much liked his comment when he said, ‘I am delighted to see 16 Welshmen in the side, but in my mind from now on they cease to be Welsh, they are British.’

⁵ The Welsh rugby player Philip Bennett (born 1948) was then the Lions’ captain.

5-: From a report from New Zealand, by Chris Lander, published in the Daily Mirror (London, England) of Friday 17^th June 1977:

WELSH ‘TAFIA’ GET CONTRACT

British Rugby looks to Wales to salvage the Lions’ pride here tomorrow.
Never can the Lions have gone into a first Test match relying so much on so many key Welshmen.
There was bitter criticism around Britain when the Welsh got 16 places on the tour. Now, with nine men in the first Test, they have the ideal opportunity to silence the anti-Welsh brigade once and for all.
There will be an almighty responsibility on the “Welsh Tafia” as they have become affectionately known on this tour.