The British- and Irish-English slang expression blue funk denotes a state of extreme nervousness or dread.

This expression occurs, for example, in A log, the newspaper column of the Irish playwright and television-writer Hugh Leonard (1926-2009), published in the Sunday Independent (Dublin, County Dublin, Ireland) of Sunday 24^th September 2000:

It was in the winter of 1961 that I flew to Manchester and first saw the studios of Granada Television where I was to work as one of the four staff writers. A fat, balding executive named Harry Elton showed me around, and it was agreed that I would travel to Manchester every Monday morning and home again on the Thursday afternoon.
Flying holds no fears whatever for me; on the contrary, what puts me into a blue funk is the thought of crashing. Nonetheless, I signed the contract.

In early American-English usage of blue funk, this expression was used in the British-English sense of a state of extreme nervousness or dread, and was frequently referred to as being of British-English origin. The following, for example, is from The People at the Polls, published in the Springfield Daily Republican (Springfield, Massachusetts, USA) of Tuesday 3^rd November 1874:

Seriousness is one thing and panic another. The average American voter is a being of weaker nerves and greater gullibility than he has been generally supposed, if he allows himself to be thrown into a “blue funk,” as our English cousins phrase it, by the alarmist talk with which interested politicians and their newspaper attachments are trying to scare him into a surrender of his private judgement and his inalienable right of thinking and choosing for himself.

—Cf. ‘blue funk’ (American usage).

In the British- and Irish-English expression blue funk:
– the adjective blue is an intensifier—as in the expressions blue blazes and once in a blue moon;
– the noun funk denotes a state of extreme nervousness or dread.

The following, about the adjective blue used as an intensifier, is from Some Slang Phrases, published in All the Year Round. A Weekly Journal. Conducted by Charles Dickens (London, England) of Saturday 9^th June 1888:

“Blue” is a favourite adjective in slang phrases. Schoolboys, in their own choice dialect, talk of “blue fear” and “blue funk.” The indefinite period known as “once in a blue moon” is a favourite with Miss Braddon, ¹ if one may judge by her frequent use of the expression. The moon will doubtless not be blue until the Greek Calends, or, as they say in Ireland, till “Tib’s Eve,” ² whenever that may be.

¹ This refers to the British novelist Mary Elizabeth Braddon (1835-1915).
² The Irish-English phrase (on) Tib’s Eve means never.

The noun funk originated in the slang of the University of Oxford, Oxfordshire, England. It is first recorded in Etymologicum Anglicanum (Oxford, 1743), by the German philologist Franciscus Junius (1589-1677), edited by the English philologist Edward Lye (1694-1767):

Funk vox Academicis Oxon. familiaris. to be in a funk. vett. Flandris fonck est Turba, perturbatio. in de fonck siin, Turbari, tumultuari, in perturbatione versari.
     translation:
Funk is a word familiar to the academics of Oxford. to be in a funk. In Old Flemish fonck is turmoil, agitation. in de fonck siin, to be agitated, to be in tumult, to be in a state of perturbation.

This entry from Etymologicum Anglicanum indicates that the English noun funk is perhaps from the Dutch noun (of unknown origin) fonk, earlier fonck, used in the phrase in de fonck, meaning: in turmoil, in difficulties, in dire straits.

The earliest occurrences of the expression blue funk that I have found are as follows, in chronological order:

1-: From The Phrenologist’s Daughter. A Tale (London: Hope & Co., 1854), by an unknown author:

Reader! have you ever been in a funk? If so, have you ever been in a blue funk? I am very sorry. ’Tis a low word, a very low word. But there is no other to express the idea, and under such circumstances what is to be done? “Fright” does not picture the feeling alluded to, no more does “alarm,” nor “terror.” No! “Funk” is the only word that will do.
Stapleton was in a blue funk.

2-: From Under the Sea, an unsigned short story published in Household Words. A Weekly Journal. Conducted by Charles Dickens (London, England) of Saturday 23^rd June 1855:

Once upon a time I persuaded Mr. Headfurst to let me accompany him on one of his submarine visits to the great three-decker which I first spoke of as sunken opposite. I was in a flutter of fright and joy such as youths who have only been down in the bell at the Polytechnic can form no idea of. I had the perfectest confidence in the machine, and, above all, in my friend Thomas, but still I was in a greater state of “blue funk” than most boys of fifteen have ever any reason to be.

3-: From Tom Brown’s School Days ³ (Cambridge: Macmillan & Co., 1857), by the British social reformer and children’s writer Thomas Hughes (1822-1896):

“Tom,” said he, “blest if you ain’t the best old fellow ever was—I do like to see you go into a thing. Hang it, I wish I could take things as you do—but I never can get higher than a joke. Everything’s a joke. If I was going to be flogged next minute, I should be in a blue funk, but I couldn’t help laughing at it for the life of me.”

³ This book was also published under the title School Days at Rugby (Boston (Massachusetts): Ticknor and Fields, 1857).

4-: From Tom Brown at Oxford (Cambridge: Macmillan & Co., 1861), by the British social reformer and children’s writer Thomas Hughes (1822-1896):

“Fear never made a man do a right action,” he summed up to himself; “so here I stop, come what may of it. I think I’ve seen the worst of it now. I was in a real blue funk, and no mistake. Let’s see, wasn’t I laughing this morning at the watcher who didn’t like passing a night by the river? Well, he has got the laugh of me now, if he only knew it. I’ve learnt one lesson to-night at any rate; I don’t think I shall ever be very hard on cowards again.”

5-: From the “account, by a gentleman present, of the late attack upon the British Legation in Japan”, published in The Bombay Gazette (Bombay, Maharashtra, India) of Thursday 12^th September 1861:

Revolvers are the order of the day, and “blue funk” prevalent.

6-: From the review of a sketch of the proceedings of an English commission ‘de lunatico’, published in La Revue des Deux Mondes (Paris, France)—review published in The Saturday Review of Politics, Literature, Science, and Art (London, England) of Saturday 23^rd November 1861:

We are conducted to the hotel where the jury is empanelled. We encounter, outside the Court, the miserable Dr. Blanding in what is called—in English, unknown, perhaps, to the Revue—a blue funk, and endeavouring to obtain from un jeune avocat irlandais some advice as to how he should support the terrible ordeal of cross-examination.