This is actually a pretty fortuitous necro-post, since after 25th and JR came out on Good Old Games, I was so perturbed by it being based on the DOS version of Judgement Rites compared to the Mac port I remembered, I installed a Mac OS 9 emulator to try and play it, with the intention of recording a playthrough so YouTube would have a copy of the good version for posterity. There were a few main flaws with the DOS version compared to the Mac port I'd played originally— The cutscene movies looked worse (and at least one, a brief prologue, was missing entirely), the transporter had the wrong sound effect (as mentioned above; this had been corrected in the Mac version), and the music sounded different (and in my opinion, worse). Completely slipped my mind that I was going to do that. I should write it down.
Anyway, while I was looking into that, I was also able to access the original Midi files and play them through the stock Mac OS X interpreter. It didn't sound the same as it did in-game, obviously, and it seemed like a lot of work that no one would care about to extract them all, so I put them aside. It took some digging this afternoon to rediscover how I'd done it (turns out, there are a couple modern tools for those of us that need to hack into obsolete Mac files), but I cracked it. And now seeing that there's interest in the community in the raw music files, I went to the trouble of extracting the whole set from both games. I've included all the files, straight from the games, including some that seem to be special-purpose background loops that use sound effects from the app rather than "real" instruments. I can provide the source sound files, as well, if anyone is interested, but they're just single instrument notes, stock '90s video game sounds, and stock TOS sounds. I can see how they might be useful for people who are more into audio, but I can also see how they wouldn't be half as interesting as the midi files.
So, without further adieu, courtesy of the original MacPlay 1990s ports of the games and old-school Mac power-users, here are the original soundtracks for Star Trek: 25th Anniversary and Star Trek: Judgement Rites. Download and do as you will. Please don't credit me I've also attached it to the post, for safety.
The JR tracks sound much better than the 25th anniversary ones, even where the tracks seem to be the same music.
Personally, I'd love it if we could put our heads together and figure out the breakdown of adapted TOS tracks versus original music, matching as many files as we can to the big La La Land TOS soundtrack. I'll try to give it a start, but I'm not as familiar as I'd like with all 600-some tracks of TOS music.

It looks like most of the original material was composed for 25th, and the additional music in JR is all straight adaptations from the TOS soundtracks.
25th Anniversary Soundtrack
00128 "title" — TOS Main Titles
00129 "danger" — “Fesarius” from “The Corbomite Maneuver”
00130 "bad1" — Romulan or "Blackship" theme from "Balance of Terror" or "Mirror, Mirror"
00131 "bad2" — From the show
00132 "funny" — Riff on Finnegan's theme from "Shore Leave"
00133 "sta43.v4" — Starts off as "The Sneaky Commodore" from "The Doomsday Machine," then original(?)
00134 "title 2" — TOS Main Titles
00135 "Sta44v5" — High-energy riff on the main title
00136 "sta45.v4" — Starts off as "Goodbye M. Decker" from "The Doomsday Machine," then original(?)
00137 "sta46v4" — Space battle victory music. May be a main theme riff from the show
00138 "sta47v4" — Space battle defeat music, original(?)
00139 "CB_Tag1" — Original(?)
00140 "cntag2" — Original(?)
00141 "CN_tag3" — Level score music(?) Original(?)
00142 "CN_TAG5" — Level score music(?) Original(?)
00143 "CW_Tag1" — Level score music(?) Riff on Main Title
00144 "CW_tag2" — Level score music(?) Riff on Main Title
00145 "CW_Tag3" — Level score music(?) Riff on Main Title
00146 "tittag2" — Level title music. Riff on Main Title
00147 "CN_TAG1" — Riff on Main Title
00148 "CB_Tag2" — Riff on Doomsday Machine theme
00150 "Big Door" — Sound Effects Loop(?)
00151 "Crane" — Sound Effects Loop(?)
00152 "Electric" — Sound Effects Loop(?)
00153 "Machine" — Sound Effects Loop(?)
00154 "Thunder" — Sound Effects Loop(?)
00155 "Main Screen" — Sound Effects Loop(?)
Judgement Rites Soundtrack
01000 "SGBatlWl" — High-energy riff on the main title; Improved version of 00135 "Sta44v5"
01001 "SGBatlNr" — Starts off as "The Sneaky Commodore" from "The Doomsday Machine," then original(?); improved version of 00133 "sta43.v4"
01002 "SGBatlBd" — Starts off as "Goodbye M. Decker" from "The Doomsday Machine," then original(?); Improved version of 00136 "sta45.v4"
01003 "SGACTN1" — Riff on Doomsday Machine theme; Improved version of 00148 "CB_Tag2"
01004 "SGACTN2" — Improved version of 00140 "cntag2"
01005 "SGACTN3" — Improved version of 00144 "CW_tag2"
01006 "SGWALK1" — Improved version of 00139 "CB_Tag1"
01007 "SGWALK2" — Riff on Main Title; Improved version of 00147 "CN_TAG1"(?)
01008 "SGWALK3" — Tense riff on Main Title; May in included in one of the longer 25th tracks
01009 "SGFAIL" — Improved version of 00141 "CN_tag3"
01010 "SGWarpTg" — Riff on Main Title; Improved version of 00143 "CW_Tag1"
01011 "SGWin1" — Riff on Main Title; Improved version of 00145 "CW_Tag3"
01012 "SGBrass1" — Riff on Main Title; May be from show
01013 "SGBrass2" — Original(?)
01014 "SGClue" — “A Clue” from “Shore Leave” (this is getting easier)
01015 "SGContr" — “Contrary Order” from “Amok Time”
01016 "SGJoust" — “Joust” from “Shore Leave”
01017 "SGOldenG" — “Old English” from “Shore Leave”
01018 "SGOutOfC" — “Out of Control/Lurch Time/Punchy Kid” from “The Naked Time”
01019 "SGParty1" — “Party Time” from “The Naked Time”
01020 "SGParty2" — “Party Time” from “The Naked Time”
01021 "SGRitul1" — “Ritual” from “Amok Time”
01022 "SGRitul2" — “Ritual” from “Amok Time”
01023 "SGRuth" — “2nd Ruth (Ruth)” from “Shore Leave” (the OS X Midi is really unkind to this one)
01024 "SGRuthL" — “2nd Ruth (Ruth)” from “Shore Leave”
01025 "SGDangr1" — “Fesarius” from “The Corbomite Maneuver”; Improved version of 00129 "danger"
01026 "SGBad1" — Romulan or "Blackship" theme from "Balance of Terror" or "Mirror, Mirror"; Improved version of 00130 "bad1"
01027 "SGBad2" — Still can’t remember which episode this was from, but, you guessed it; Improved version of 00131 "bad2"
01028 "SGTitle" — TOS Main Title
01029 "SGTitTg" — Level title music. Riff on main theme; Improved version of 00146 "tittag2"
01030 "SGFunny1" — Riff on Finnegan's theme from "Shore Leave"; Improved version of 00132 "funny"
01031 "SGDedEmy" — Space battle victory music. May be a main theme riff from the show. Improved version of 00137 "sta46v4"
01032 "SGDedYou" — Space battle defeat music, original(?); Improved version of 00138 "sta47v4"
01033 "SGThelo1" — “Prime Specimen” from “The Cage”
01034 "SGThelo2" — Talosian theme from “The Cage,” not sure which track
01035 "SGThelo3" — From the show(?)
01036 "SGThelo4" — From the show(?)
01037 "SGThelo5" — From the show (?)
01038 "SGThel51" — From the show(?); similar to 01037 "SGThelo5"
01039 "SGThel52" — From the show(?)
01040 "SGTime" — “Time Reverse” from “The Naked Time”
01041 "Bridge Bkgnd" — Sound Effects Loop(?)