I've just done a quick search on Roud for Source:Naval Songs and Ballads and Roud No:Not empty and there are 64 results in that list. These are the ones with non-broadside sources and are therefore most likely to have tunes available. I haven't checked the Roud numbers for which might have tunes.

Many of the tunes in Firth from broadside sources specify tunes and I can check for those in Simpson to see which tunes are known (this obviously excludes the like of To a pleasant tune).

The list below removes duplicates from the Roud search, and are the most likely ones to find sung. There may well be others from a text only source that people have put tunes to.

Perhaps Sam can tell us what material he is looking for. Does he want commercial recordings? Does he want the exact version as it appears in Firth? If he doesn't mind non-commercial recordings, it's quite likely that people here could do that using existing tunes from other versions, or the broadside tunes where they exists. (I haven't time to check broadside tunes at the moment, but I can do it tomorrow probably).

Mick

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ADMIRAL BENBOW Roud number        3141
BATTLE OF THE SHANNON AND CHESAPEAKE Roud number        963
BILLY TAYLOR (A BURLESQUE BALLAD) Roud number        158        
BLANCHE FRIGATE, THE Roud number        4583
BORNEO HEROES, THE Roud number        V1666
BRAGS OF WASHINGTON, THE Roud number        22795        
CAPTAIN BARTON'S DISTRESS ON BOARD THE LICHFIELD Roud number        V1856
CAPTAIN COBB'S BRAVERY Roud number        7005        
CAPTAIN KID'S FAREWEL TO THE SEAS; or, THE FAMOUS PIRATE'S LAMENT Roud number        1900        
CAPTAIN MANSFIELD'S FIGHT WITH THE TURKES AT SEA Roud number        951
CAWSAND BAY Roud number        22827        
CONSTITUTION AND GUERRIERE, THE Roud number        626        
DEATH OF ADMIRAL BENBOW, THE Roud number        227        
DEATH OF PARKER, THE Roud number        1032        
DOWNFALL OF PIRACY, THE Roud number        8115        
ENGLAND'S TRIUMPH AND HOLLAND'S DOWNFALL; or, THE SECOND ROYAL VICTORY Roud number        V1307        
FAIR SALLY LOV'D A BONNY SAILOR Roud number        V2837
FAMOUS SEA-FIGHT BETWEEN CAPTAIN WARD AND THE RAINBOW Roud number        224        
FANCY FRIGATE, THE Roud number        2563        
FEMALE LIEUTENANT, THE; or, FAITHLESS LOVER REWARDED Roud number        158        
HAWKE'S ENGAGEMENT Roud number        18256        
HEARTS OF OAK Roud number        V391        
HOSIER'S GHOST Roud number        22377        
HUMOURS OF THE ROYAL BILLY, THE Roud number        V3972        
JOHN DORY Roud number        249        
LOSS OF THE CENTAUR, THE Roud number        V851
MAIDEN'S FROLICK, THE; or, A BRIEF RELATION HOW SIX LUSTY LASSES HAS PREST FULL FOURTEEN TAYLORS... Roud number        V5616
NEWS FROM THE COAST OF SPAIN; or, A TRUE RELATION OF A BRISK AND BLOODY ENCOUNTER... Roud number        V1994        
NEW SONG COMPOSED BY THE WOUNDED TARS AT THE SIEGE OF BOULOGNE, A Roud number        3175
OH CRUEL (ANSWER TO) Roud number        V1837        
OH CRUEL Roud number        V1836
PAUL JONES Roud number        967        
ROYAL TRIUMPH, THE; or, THE UNSPEAKABLE JOY OF THE THREE KINGDOMS Roud number        V1306        
RUSSIANS WON'T COME OUT, THE Roud number        V5353        
SAILOR'S ONELY DELIGHT, THE Roud number        134        
SAUCY SCYLLA, THE Roud number        V4653
SHANNON AND CHESAPEAKE, THE Roud number        1583
SIR ANDREW BARTON Roud number        104        
SLAVE CHASE, THE Roud number        22376        
TERRIBLE PRIVATEER, THE Roud number        9381        
YE PARLIAMENT OF ENGLAND Roud number        2078