The OED's citation of Nathan Bailey, Dictionarium Britannicum (1730), mentioned in the posted question, seems to be off by one edition and six years. The first edition of Bailey's dictionary has no entry between

To BLUBBER, to cry or foul the cheeks with tears.

and

BLUE a colour well known.

In contrast, the second edition of Dictionarium Britannicum (1736) has this entry between "BLUBBER LIP'D" and "BLUE":

BLUDGEON, an oaken stick or club.

About 20 years later, Samuel Johnson, A Dictionary of the English Language (1755) has this entry:

BLUDGEON n. s. A short stick, with one end loaded, used as an offensive weapon.

Although the OED seems to have misidentified the year of the first dictionary entry for bludgeon, I found two instances of bludgeon that really are from 1730. From The Grub-Street Journal (January 8, 1730):

Tuesday, Jan 6 ... Last Sunday Night about eleven o'Clock, Mr Stalton the brewer, was attack'd neat the end of Thrift street, Sobo, by a single Vilain, who pul'd out a Bludgeon, and knock'd him down, and then rifled him of his Gold watch and Money, which Mr Stalton believes to be upward of 12 l and then made off, and yesterday he was seized offering Mr Stalton's Watch to pawn, and being carried before a Magistrate, was committed to New-Prison.

And from The Political State of Great-Britain (March 1730):

The wicked People that carry on this inhuman Trade of knocking People down thus, before they rob them, or so much as ask them for their Money, are observed to do it with an Iron Bar or Truncheon, which they call a Bludgeon ; which is about two Foot in Length, and is carried under their Cloaths, by which they give such a desperate Blow on the Head, as several have had their Skulls broken with them, and have never recover'd the Injury of it; and it is observable also, that the Women carry these cruel Weapons about them as well as the Men, and perhaps use them the more cruelly also of the two.

This last instance suggests that bludgeon started out in English as street slang; however, it doesn't appear in any of the dictionaries of English slang from the eighteenth and early nineteenth centuries that I consulted.

The OED's citation of Nathan Bailey, Dictionarium Britannicum (1730), mentioned in the posted question, seems to be off by one edition and six years. I checked my print edition of the full OED, and it does indeed cite the 1730 edition of Bailey's dictionary. But that edition of Bailey's dictionary has no entry between

To BLUBBER, to cry or foul the cheeks with tears.

and

BLUE a colour well known.

In contrast, the second edition of Dictionarium Britannicum (1736) has this entry between "BLUBBER LIP'D" and "BLUE":

BLUDGEON, an oaken stick or club.

About 20 years later, Samuel Johnson, A Dictionary of the English Language (1755) has this entry:

BLUDGEON n. s. A short stick, with one end loaded, used as an offensive weapon.

Although the OED seems to have misidentified the year of the first dictionary entry for bludgeon, I found two instances of bludgeon in the wild that really are from 1730. From The Grub-Street Journal (January 8, 1730):

Tuesday, Jan 6 ... Last Sunday Night about eleven o'Clock, Mr Stalton the brewer, was attack'd neat the end of Thrift street, Sobo, by a single Vilain, who pul'd out a Bludgeon, and knock'd him down, and then rifled him of his Gold watch and Money, which Mr Stalton believes to be upward of 12 l and then made off, and yesterday he was seized offering Mr Stalton's Watch to pawn, and being carried before a Magistrate, was committed to New-Prison.

And from The Political State of Great-Britain (March 1730):

The wicked People that carry on this inhuman Trade of knocking People down thus, before they rob them, or so much as ask them for their Money, are observed to do it with an Iron Bar or Truncheon, which they call a Bludgeon ; which is about two Foot in Length, and is carried under their Cloaths, by which they give such a desperate Blow on the Head, as several have had their Skulls broken with them, and have never recover'd the Injury of it; and it is observable also, that the Women carry these cruel Weapons about them as well as the Men, and perhaps use them the more cruelly also of the two.

This last instance suggests that bludgeon may have originated in English as street slang; however, it doesn't appear in any of the dictionaries of English slang from the eighteenth and early nineteenth centuries that I consulted.