The song, titled "The Mermaid" and sung by Sarah Walker and Thomas Allen, is on The Sea (Hyperion CDA66165; track 12; with sound clip).

There's another set of lyrics in The Scottish Students' Songbook (Bayley & Ferguson, 1897 edition, pp. 120-121; with tune).

            THE MERMAID
    (Words by A.J.C.; Arr. by Michael Watson)

               1.
Oh, 'twas in the broad Atlantic,
'Mid the equinoctial gales,
That a young fellow fell overboard
Among the sharks and whales.
And down he went like a streak of light,
So quickly down went he,
Until he came to a mermaid
At the bottom of the deep blue sea.
Chorus: Singing, Rule Britannia, Britannia, rule the waves!
And Britons never, never, never shall be married to a mermaid
At the bottom of the deep blue sea.

               2.
She raised herself on her beautiful tail,
And gave him her soft, wet hand,
"I've long been waiting for you, my dear,
Now welcome safe to land.
Go back to your messmates for the last time,
And tell them all from me,
That you're mar-ri-ed to a mer-ma-id
At the bottom of the deep blue sea."
Chorus: Singing, Rule, Britannia, etc.

               3.
We sent a boat to look for him,
Expecting to find his corpse,
When up he came with a bang and a shout,
And a voice sepulchrally hoarse.
My comrades and my messmates,
Oh do not look for me,
For I'm mar-ri-ed to a mer-ma-id
At the bottom of the deep blue sea.
Chorus: Singing, Rule, Britannia, etc.

               4.
In my chest you'll find my half-year's wage,
Likewise a lock of hair,
This locket from my neck you'll take,
And bear to my young wife dear.
My carte-de-visite to my grandmother take,
Tell her not to weep for me,
For I'm mar-ri-ed to a mer-ma-id
At the bottom of the deep blue sea.
Chorus: Singing, Rule, Britannia, etc.

               5.
The anchor was weighed, and the sails unfurled,
And the ship was sailing free,
When up we went to our cap-i-taine,
And our tale we told to he.
The captain went to the old ship's side,
And out loud bellowed he,
"Be as happy as you can, with your wife, my man,
At the bottom of the deep blue sea."
Chorus: Singing, Rule, Britannia, etc.

~Masato