Here's a rough translation of most of it. There may be errors, and I would be happy for more fluent speakers of proper deep Yorkshire to correct them.

    "Come on, son. Gerrup! Ah've a surprise for thi!"

Come on son, get up! I've a surprise for you!

    "Ah see Leetning lost three on t' thutty-niners
     t' other week, Poke. Bloody belt'll kill some'dy
     sooin, tha knows."

I see Lightning lost three (fingers) on the thirty-niners the other week, Poke. Bloody (conveyer) belt will kill somebody soon, you know.

Poke was the author's father's nickname.

"Lightning" was his father's best friend's nickname.

"Thutty-nine" was the coalface they were working on.

    Dad often threw a sickie ...

To take a day off as sick leave, despite not being ill.

    "Stick wi' me, son. Tha'll be reet."

Stick with me, son. You'll be right."

    "Hey up, Poke. Is that thy lad?"

Hey up, Poke. Is that your son?

    "Hey up, Young Pokey. Is tha barn darn t' pit?"

Hey up, son of Poke. Is the child down the pit?

    "Aye, sither. Ah'm barn darn t' thutty-niners
     wi' t' fa'ther."

Aye, you see. I'm (the) child down the coalface with the father.

    "He's a cheyky young bleeder. Tha wants to
     gi'im thick end o' thi belt."

He's a cheeky you lad. You want to give him (the) thick end of your belt.

    "Tha knows Leetning, dun't tha?"

You know Lightning, don't you?

    "Aye, sither, but even then Ah still do bart
     twice as much as thi fa'ther."

Aye, you see, but even then I still do about twice as much as your father.

    "But thy an't had thi surprise yet,"

But you haven't had your surprise yet.

    "Poke's lad darn yet?"

(Is) Poke's son down (here) yet?

    "Reet oh. Ah'll start droppin 'em.
     Watch thi'sens."

Right oh. I'll start dropping them. Watch yourselves.

    "Tha wain't say nowt to thi mam nar,
     will tha? This is just between us men."

You won't say anything (lit. nothing) to your mother, will you? This is just between us men.

    "But tha knows why he done it, dun't tha?"

But you know why he did it, don't you?

    "Nay, lad,"

No, lad.

    "Thi dad had to grease a few palms to
     get thi darn t' pit that day, tha knows."

Your dad had to bribe a few people to get you down the pit that day, you know.

    "Nar look at thi. Tha passed thi Eleven Plus, ...

Now look at you. You passed your 11-plus (a significant school exam),

    "... tha's bin to college an' tha's got
     a reet good job, an't tha?

... you've been to college, and you've got a right good job, haven't you?

    "Exactly! Sither nar, sunshine?"

Exactly! See it now, sunshine?