There's another Scottish song that uses this tune, "Will i the Lum".

Here are two recordings of the song, it appears to be quite rare:
First:
http://tobarandualchais.co.uk/en/fullrecord/39412
Second:
https://sounds.bl.uk/World-and-traditional-music/Peter-Kennedy-Collection/025M-C

The song starts at 1:04 in the recording. The first song is "Wee Cooper o' Fife", after that comes "Will i the Lum" whith the beautiful Down, Derry Down chorus.

It is a variant of the song that is known as "Will the Weaver" (Roud No. 432) in the US.

I've been searching for songs that make use of this tune quite extensively, but there are not many recent recordings of it from Britain as far as I know.

Here are two fragments of Well Sold The Cow sung to this tune, also from Scotland:
First
http://tobarandualchais.co.uk/en/fullrecord/37334
Second:
http://tobarandualchais.co.uk/en/fullrecord/19970

Other than that I only know of Bob Lewis' song, "The Cobbler", as sung on "The Painfull Plough" (2003) and also on "Two Bobs' Worth (MTCD374)" (2017).

A. L. Lloyd sang his arrangement of "The Old Bachelor" from Greig-Duncan to the tune, but you never know with Lloyd.

Finally, Ewan MacColl sung "The Coal-Owner and the Pitman's Wife" to this tune, but I don't think there is much traditional about this song.

Apart from that I know of several songs recorded in Canada and the Adirondack region that use the tune.

King John and The Bishop comes to mind. There's a splendid, but sadly poorly sounding recording of it in the Flanders Collection:
https://ia801209.us.archive.org/32/items/HHFBC_tapes_D70B/D70B%20sideA.mp3

The song starts at 37:00 on the tape.

Hmm, the Irish song Drimindown, where a man laments the loss of his cow does not use the Down Derry Down chorus, but the tune is very, very similar, at least in the Canadian version recorded by Helen Creighton from Mr. Ernest Sellick.

You should be able to listen to the recording on youtube:
https://www.youtube.com/watch?v=uUNK1rIKUtw

Here's a fragment of John Hopper's Hill sung by Warde Ford of Wisconsin:
https://www.loc.gov/item/2017701449/

"A Comical Ditty", sung by Almeda Riddle on "More Ballads and Hymns from the Ozarks" (1976) also uses a form of the tune and chorus, but it sounds rather different, I've uploaded the recording to youtube so you can give it a listen for yourself:
https://www.youtube.com/watch?v=KeVIRLBYA3w

You already know about Beaver River and Miner Hill, as sung by Ted Ashlaw. I was going to link the recordings, but sadly, the website adirondackmusic.org is no longer working for me.

There's also "The E. C. Roberts / Red Iron Ore" as sung by Stanley Baby on "Songs of the Great Lakes" (1964):
https://www.youtube.com/watch?v=inVD1Vg9_uA

The Ballad of Blue Mountain Lake, or Loon Lake also uses the tune, various recordings of it are in the Flanders collection.

Also "The Aviator's Hymn", as sung by Yankee John Galusha, another Adirondack singer.

Some versions of "Little Brown Bulls" also use the Derry Down chorus, but the tune sounds different to me.

And there is also a quite strange song in the Sidney Robertson Cowel Wisconsin Folksong Collection called The Pickled Jew or The Barrel of Pork that uses the tune:
https://search.library.wisc.edu/digital/AVDTDHMVLMQ2DW8T

The humour of the song is in bad taste from a modern point of view, but the tune as sung by Robert Walker is splendid.

Anyways, those are all I could think of on the top of my head. I'm sure there are many more rather recent songs in Canada/Northern US that use the tune as it was very popular in the lumber camps.