This is pretty close to the way that Blue Murder sings it:

CHORUS:
Help me, Bob, I'm bully in the alley,
Way, hey, bully in the alley!
Help me, Bob, I'm bully in the alley,
Bully down in shinbone al!

Well, Sally is the girl down that I love dearly,
Way, hey, bully in the alley!
Sally is the girl that I spliced nearly.
Bully down in shinbone al!

CHORUS

For seven long years I courted Sally,
Way, hey, bully in the alley!
All she did was dilly-dally.
Bully down in shinbone al!

CHORUS

I'll leave Sal and I'll become a sailor,
Way, hey, bully in the alley!
I'll leave Sal and ship aboard a whaler.
Bully down in shinbone al!

CHORUS

I'll come back and I'll marry Sally,
Way, hey, bully in the alley!
We'll have kids and count them by the tally.
Bully down in shinbone al!