Updating the list once more as I've found yet another track, "Music for Percussion 14-19", composed by Laurie Johnson and from KPM. Specifically, track 17, "Nursery", was used in "The Villain Pursues Her".

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Updating as somebody from the Production Music Wiki ID'd a couple of Themes International tracks as being used in these cartoons:
Jingle 4b - Alan Parker (Heard at the end of "Ain't Nature Grand")
Quickie 3 - James Clarke (Heard at the beginning of "Country Boy Rabbit".)

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Thanks to the Internet Archive's ongoing 78 RPM preservation project uploading some old Conroy 78s among other goodies within the past few days, I was able to finally confirm one of my holy grails of the R&TVP stock cues, the sax music Felix plays at the end of "Mr. Do-All". It's "Sax Rock" by Reg Owen, from Conroy BM 260.

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Found the tracks that open and end "Lumberjack" on album BM 194 from the Conroy library:
"Wee Willie" (title card)
"Happy Time" (ending)

Both pieces are composed by Bruce Campbell.

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May as well drop some theories I have with some of the currently unidentified tracks (the ones I'm most interested in finding someday specifically):

The music heard throughout "Art for Art's Sake" and "The Steadfast Tin Soldier" = These tracks are, to borrow a phrase from "The King and I", a puzzlement. The more timeless sounding orchestral cues heard throughout fit the on-screen action fairly well, but they're most likely expertly edited together from an unidentified stock library. I have my suspicions some of the "Tin Soldier" cues are Harry Lubin tracks from the Harrose library (which was distributed by Emil Ascher during the '60s and '70s), they have a similar sound to some of his light-hearted tracks found on CPM's archive library.

The comedy chase music heard in "Soda Jerks", "Misses His Swiss", and "Music and Charm" and the alternate version of this cue heard in "Tuning In": I have a hunch this is from some "tape reel-only" album from one of the usual British music libraries - it has a very English feel to it. Whatever album has it, I have a hunch the majority of the tracks heard in "On Duty" are also on there too. I've speculated that maybe these two cues were meant as a "main and end title" for the package if any stations chose to air it as a half-hour item (which could explain why they show up in a few shorts with mostly in-house music like "Soda Jerks"), but the fact one of the "On Duty" tracks shows up in a Sesame Street bit sort of debunks that.

In addition, I have a good hunch more of the sound effects featured in these shorts besides just the Sam Fonteyn percussion tracks on KPM 1047 might be on another "tape only" album that's exclusively percussion/comedy effects, especially one cut in particular that never fails to play unedited regardless if it makes sense for the scene or not (the xylophone note/slide whistle descent combo heard in "The Villain Pursues Her" when the wolf villain lifts up the stove, for instance).

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Found a short cue used in "On Duty" on the same 78 as "Traffic Patterns": "That's All" from Eric Austen, the same composer of the aforementioned piece. It's the two-note "DUN DUN!" heard just before Roxy plays an upbeat tune on the player piano, and not the tail end of the honky-tonky piano piece heard before it as previously suspected.

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Another track's been ID'd thanks to "WSBG" on the Library Music Themes forum: "Four Jokers Suite Pt. 4" by Bruce Campbell, which shows up at the beginning and end of "Accidents Won't Happen".

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Finally discovered one of my "holy grail" cues from these redrawns: "Opa's Charleston" by Rex Connor, from the Audio library (album #BMP 102) to be specific. It can be heard in "The Under Dog", and the cue itself can be heard here. 

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