The Daily Telegraph (London, England) has been nicknamed (Daily) Torygraph because of its faithful adherence to the Tory Party, i.e., the British Conservative Party.
—Cf. also ‘Tory-lite’: meanings and origin.

For example, the following occurrence is from The Unmasking Of The Iron Lady ¹, by Peter Kellner and Vincent Browne, published in The Sunday Tribune (Dublin, County Dublin, Ireland) of Sunday 19^th January 1986:

The Daily Telegraph is so loyal to the Conservative Party that among journalists it is known as the “Daily Torygraph”.

¹ The British Conservative stateswoman Margaret Hilda Thatcher (1925-2013) was Prime Minister from 1979 to 1990.

The earliest occurrences of (Daily) Torygraph that I have found are from Letters to the Editor, published in The Guardian (London and Manchester, England):

1-: On Saturday 16^th September 1978:

Lively minds in little measure
Sir,—Being 5ft. 3in. tall with an inside arm measurement of 22in., I get into the most awful mess trying to read the Guardian, especially on the Northern Line in the rush hour. Torygraph and Excess ² readers to the right and even, surprisingly, to the left take great exception to bits of the Guardian cascading over their laps when I attempt to turn the pages. Would it not be possible to publish a tabloid version for your shorter readers?
(Miss) D. E. Shaw,
London SW2.

² Here, Excess is probably a nickname for the Daily Express (London, England), a mid-market centre-right newspaper. Originally a broadsheet, the Daily Express switched to its current tabloid format in 1977.

2-: On Wednesday 8^th July 1981:

Sunday blues
Sir,—When the Times group was bought by Rupert Murdoch ³,  I decided that I could no longer justify to myself buying the Sunday Times. (I should add here that I had already seen the light and changed to the Guardian long before.) Now, thanks to the Monopolies Commission, it appears that the Observer will go a similar way and end up in the hands of Rowland and Lonrho ⁴. Consequently I will stop buying the Observer.
As there is no other Sunday paper worth looking at—this, of course, includes, the Sunday Torygraph—is the moment not ripe for a Sunday Guardian, or will I just have to sleep all through Sundays?—Yours,
Anthony Hampton.

³ Keith Rupert Murdoch (born 1931) is an Australian-born U.S. publisher and media entrepreneur. As the founder and head of News Corporation, he owns major newspapers in Australia, Britain and the USA, together with film and television companies and the publishing firm HarperCollins. (The British newspaper division of News Corporation is News International.)
⁴ Roland Walter ‘Tiny’ Rowland (1917-1998) was a British businessman, corporate raider, and Chief Executive of the London and Rhodesian Mining and Land Company Limited, later Lonrho, from 1962 to 1994. Under his leadership, the firm expanded beyond mining, and dealt in particular in newspapers—cf. meaning and origin of the British phrase the acceptable face of ——.

John Stevens, writing from London, mentioned several other nicknames for British newspapers in an article published in The Age (Melbourne, Victoria, Australia) of Wednesday 4^th February 1987:

The move to Wapping ⁵ has coincided with the continued decline in the standard of ‘The Times’, Murdoch’s British flagship. […]
The other prestige paper from the Wapping stable, the ‘Sunday Times’, is also in decline. Media commentators are saying Mr Murdoch will have to give these problems more personal attention, but he is unlikely to be losing sleep while the ‘Sun’ ⁶  continues to prosper. This week the cheeky tabloid crowed that its sales — four million daily — were a million ahead of those of its nearest rival, the ‘Daily Muckswell’, a slighting reference to Mr Robert Maxwell’s ‘Daily Mirror’ ⁷, and more than two million ahead of the ‘Daily Fail’ (‘Mail’) ⁸ and ‘Daily Distress’ (‘Express’).

⁵ In 1986, Rupert Murdoch moved his News International printing and publishing works to Wapping, a district in East London.
⁶ The Sun (London, England) is the red top (downmarket tabloid newspaper) par excellence. Founded in 1964, it switched to tabloid format upon Rupert Murdoch’s takeover of the paper in 1969.
⁷ The Daily Mirror (London, England) is the oldest British tabloid—it adopted that format in the 1930s. Robert Maxwell (Jan Ludvik Hoch – 1923-1991) was a Czech-born British publisher and media entrepreneur.
⁸ The Daily Mail (London, England) is a mid-market right-wing newspaper.

The Daily Mail has been nicknamed not only Daily Fail, but also Daily Wail—as mentioned by Peter Preston in The Observer (London, England) of Sunday 25^th February 2001 about “a vicious new war” between the Daily Mail and the Daily Express:

When the Express’s new editor, Chris Williams (an old Mail hand years before) got his job, the boot went in. […]
[…]
[…] The Daily Mail is the Daily Wail or the Daily Fail. Its agenda reeks of ‘bitterness and negativity’, its columns resound to the ‘rantings of Little Englanders’.

Another nickname for the Daily Mail is Daily Heil—as explained by Ben Bromley in The Chippewa Herald (Chippewa Falls, Wisconsin, USA) of Tuesday 23^rd January 2018:

Many nicknames strike at editorial boards’ politics. In England, the Daily Mail is punished for supporting Fascists in the 1930s through its nickname, the “Daily Heil.”

In this article, about the nicknames given to newspapers in general (not only to British newspapers), Ben Bromley also wrote:

“The Sun” becomes “The Scum.”

And, indeed, the British red top The Sun has been nicknamed The Scum.

Cf. also:
– 19th-century nicknames for London newspapers
– origin of Grauniad (colloquial name for The Guardian)
– British Newspapers Online