The phrase like billy-o, also like billy-oh, means very much, very intensely.

Of unknown origin, the word billy-o(h) occurs only in this phrase. It is apparently composed of Billy, pet form of the male forename William, and the suffix -o, used to form slang and colloquial nouns, adjectives and interjections.
—Cf.:
– the phrase to lie doggo (1882);
– the noun beano (1883), abbreviation of beanfeast;
– perhaps the adjective blotto (1917), meaning drunk, said to be from the noun blot.

These are the earliest occurrences of like billy-o(h) that I have found, in chronological order:

1-: From Aberbeeg. A Tramp in the Western Valleys, by ‘Tramp’, published in The Star of Gwent (Newport, Monmouthshire, Wales) of Saturday 11^th August 1860—Aberbeeg and Abertillery are in Monmouthshire, Wales:

I saw in the distance smoke arising from some works […]. A few more paces brought me to the Abertilery tin works, where I could soon perceive, by the sulphuretted atmosphere, that the commodities used for manufacturing tin were different from those used in making iron. Having some knowledge of the making of iron, I, of course, felt anxious to have a peep into the secrets of tin making, so I accosted a well-dressed and apparently respectable man, who wore spectacles, “Are strangers allowed to visit the various departments in this forge, without being considered intruders, sir?” “You are quite at liberty, sir, to walk through, and inspect the whole works,” was the kind and gentlemanly reply. Sticking to my first impression of seeing all I could, I began at the west end of the forge, where pig iron was taken through the first process in tin-making. After looking for some time at a poor fellow, who appeared to be half roasted before his puddling furnace, with a nose quite as rosy as any determined grog drinker in the country, I inquired of a lad what he was doing? “Dividing his charge into six balls, sir,” was the reply. “What branch of tin making is that?” said I. “He is a puddler, sir.” “Well, puddling is very hard work, and very hot work, is it not?” “Yes sir,” said my informant, “that man works very hard all the seven days. His name is Daniel, and he is a preacher; he works 12 hours a-day at this work, and walks long journeys and preaches twice almost every Sunday.” “Have you ever heard him preach?” “Oh yes, sir, many times.” “What sort of a tool is he?” “Why, there are many worse, sir: he begins rather slowly, but he’ll go on like billyo after he once gets warm.”

2-: From The Baron de Bonchose at Hendon and Lord’s, by ‘A. H. B.’, published in the Sporting Gazette (London, England) of Saturday 13^th July 1867:

Encore in ze evening, I go vunce more me to ze Velsh Harp. I like ze bang, bang, of ze gun, and ze brave boy vat put les birds in les trappes, and duck ze head of him, vhile les shootair go off. It vas only for les sweepstake, and Sir Fred Johnstone display vunderful quick at les pigeons, he turn zem ovare and ovare; he suap zem vun altare ze odare like Billy oh!

3-: From an article about a fire at Chawleigh, published in The Western Times (Exeter, Devon, England) of Friday 27^th August 1869:

This disastrous fire broke out about two o’clock on Wednesday afternoon in a small building in the centre of the village. […] Mr. Webber, blacksmith, saw about the time stated smoke and presently fire issuing from the slated roof of this building, and he immediately exclaimed to some one near “That place is afire, for God’s sake let us try and confine it there.” “Yes,” said a bye-stander, a labourer, “and I zee’d et too comin’ out like Billy O.” The blacksmith and “Billy O” and hundreds of other people used their utmost efforts to stop it.

4-: From County Petty Sessions, published in The Western Gazette and Flying Post (Yeovil, Somerset, England) of Friday 1^st April 1870:

Richard Groves, of Bruton, a lusty young fellow, was summoned by a motherly-looking lady named Ann Spratling, for assaulting her on the night of March 19. […] Defendant denied that he shook his fist at complainant or threatened to dip her; but, with a sad want of gentlemanly gallantry, he assured the Bench that she was — well not sober; and added “She cumm’d out swearing at I like Billy o!”

5-: From Judy, or the London Serio-Comic Journal (London, England) of Wednesday 15^th June 1870:

At the sitting of the Council at Rome, on Friday last, Bishop Maret is reported to have been “shamefully interrupted by Cardinal Bilio.”—Query: Did he use Billingsgate like Billy, oh!

6-: From Fishing. Thames Angling, published in The Sporting Gazette (London, England) of Saturday 5^th September 1874:

“Our angler […] made one vigorous effort to land the fish, broke his tackle, and lost him.”
“I should a jumped in and a collared un,” interrupted Aquarius; “we ’ad a dog as ud a fetch’d un out like Billy oh.”

7-: From Spalding Petty Sessions, published in The Lincolnshire, Boston, and Spalding Free Press (Spalding, Lincolnshire, England) of Tuesday 28^th March 1876:

Henry Tooley Whitsed, farmer, of Portland, was summoned by John Howell, potato merchant, of Holbeach, for committing an assault upon him at Holbeach, on the 3rd March. […] George Westmoreland said he was an innkeeper at Holbeach and knew Howell very well. By accident he was near the Bell on the night in question, when he saw Whitsed attack Howell. Seeing the blood flow, he interfered, and took Whitsed off. He saw Gooch there, and he said “He’s got the money in his pocket, and let’s have it out of him.” He struck Mr. Whitsed because he put up his hand to strike him.—Cross-examined: Mr. Whitsed took hold of Howell and “let him have it,” and the blood commenced to flow “like billy-oh.”

8-: From Newport Police Intelligence, published in The Star of Gwent and South Wales Times (Newport, Monmouthshire, Wales) of Saturday 2^nd June 1877:

James Evans, captain of the smack Britannia, was summoned for throwing ballast into the river Usk, thereby impeding its navigation. […] William Hughes, a labourer, said he was passing down the river in a boat on Sunday night, when he saw the defendant’s smack on the East side. The boat sailed over to near where the smack was moored, and then they saw some sand coming through the port hole into the boat. Afterwards witness saw the captain “chucking the ballast up like Billy oh.”

9-: From Lokil Lettor.—No. 172, published in The Blyth Weekly News (Blyth, Northumberland, England) of Saturday 18^th August 1877—in Scottish and northern dialect, the noun keek means a peep:

Mistor Edditor, Deer Sor.—Travelling throo thi Coonty o’ Durham lately, it wis thi extreem pleashor o’ yor awd contributor ti get a keek it that seet o’ lairning an’ world-famed Educashunil Institute—Ushaw College […]. Thi students war gan away fur thor hawlidays thi week eftor we war there, an’ iverything wis consequently in confushun. Thi servent lasses wis dusting doon thi glass cases, an’ sitters, like billyo.

10-: From Cribbings from Contemporaries. Rowan Tree Castle, published in Ben Brierley’s Journal, of Literature, Science & Art (Manchester, Lancashire, England) of Saturday 16^th February 1878:

Jack found himself back again like a bad penny, and speeding on the wings of love and two sturdy legs towards the cottage where dwelt the charming Poll. There was a light in the window, it being night; so Jack thought he would have a keek before he entered. He had a look accordingly. Then Jack jumped clean off the ground, about a yard in the air.
“My eyes!” cried he; “why, bless my soul,” he said, “if there ain’t Poll herself in tow with a long white-chafted chap, with Sunday-go-to-meeting close on. And—why—yes—d—n me, if he bean’t a kissing of her.”
In ran Jack, and up rose Poll and the parson.
“Oh!” said Poll, “I is so glad you’ve come back, Jack. This here gemlem has been a converting of me.”
“Oh, I see’d it,” said Jack; “I see’d it with my own eyes. I see’d him a-kissing and a-hugging of ye like billy-o.”
“I scorn the insinuation,” said the man in black clothes, who, by the way, was a strict teetotaler, a staunch anti-tobaccoite, and an everything-that’s-good.

11-: From a letter by ‘Sairey Camp’, published in The Southend Standard, and Essex Weekly Advertiser (Southend-on-Sea, Essex, England) of Friday 20^th September 1878:

Sir,—Winter raly av come now, and no mistake. Law how that there wind did blow about 2 o’clock a Sunday, and no water carts, and the dust a blowin up like Billyoh, as the sayin is.