Liland, It was Bradypus, not me who was talking about Lingham. But as soon as I saw Bradypus's description, I knew that I know the tune & it turns out I have 2 & possibly 3 printed versions of it. I just love singing it.
Thomas Jarman's original composition was an English fuging tune published in his Sacred Music ca. 1803. It has also appeared in the US in de-fuged versions. The recording on Cyberhymnal & Bradypus' description retain the fuge & that may be why you can't make it fit.
It's also been published under the names Nativity and Desert. It's in the 1991 edition of the Sacred Harp with the tune name of Nativity, Jarman's original title for the tune. While the SH has lots of fuging tunes, the person who submitted it was only familiar with a defuged version so that's how it got into the Sacred Harp. That defuged version was in Joseph Funk's Harmonia Sacra.
In "An American Christmas Harp" (the source for most of my information) Lingham it printed with the fuge & is provided with the words:
But let us join the cheerful song,
With joy and pious mirth,
with joy and pious mithn,
And all with grateful heart and voice
Proclaim the Saviour's Birth (repeat phrase and individual as necessary to fit the part)
The compiler of the Christmas Harp found her arrangement in The American Church Harp, 1856. It also reported to be in the hymnal of The Reformed Presbyterian Church
I'd be glad to send the Sacred Harp & Christmas Harp versions to you. I have the newest ed. of Harmonia Sacra at home so I may have another version there as well.