This is now live on Paramount+ though I had to do a specific search for it as it didn't just immediately pop up.
Some quick impressions: I think this is going to be a little more controversial with some fans than people expect. I watched about the first 20 minutes. The color scheme especially when we go inside the Klingon ship at the beginning (much of the red has been removed, it plays "blacker" in hue) appears to have been heavily tweaked, especially after I watched the TMP 4K theatrical cut on UHD a week ago. It may be personal taste, but other scenes seem darker overall (Could also be the Dolby Vision HDR that Paramount+ puts out -- in fact I might have to reassess this once I see it on disc itself).
I'm also not sure I'm onboard with some of the audio alterations. The mix itself -- and it could be the streaming element and compression at play -- seemed a little "brittle" to me. Again, I'll wait for the disc to cast judgment there.
Yet some subtle but noticeable FX were added at times. Was it really necessary to add effects going over some of Goldsmith's score when Kirk and Scotty are in the shuttlecraft seeing the ship for the first time? Or when the Enterprise is ready to leave drydrock? I don't want anyone thinking these effects are overwhelming to the score or seem out of place -- but I never thought these scenes ever needed them before. The music carried it completely, which is all you needed.
Some of the remixed ADR is a mixed bag too. When Kirk gets off the shuttle in the San Francisco docking station, you can now hear the dialogue much clearer between Kirk and Sonak, but there's now next to no crowd noise in the backdrop at all. It makes it sound like they're in a confined space. Even some of the effects that didn't need tweaking -- like the sound of the transporter room accident -- have also been changed...and I don't really understand why.
Visually, the new effects work seems to be fine, but I've never had a problem with the old FX work. When the Kirk shuttle lands in SF, again, they added another shuttle on the right hand side of the screen -- but does it make the scene look more attractive? I went back to the original theatrical cut and felt that untouched matte painting functioned fine as it was. Still I haven't gone in far enough to see what else was reworked in the meat of the film, so again, I'll offer some more thoughts later.
I do like the extra footage and always have -- most of it anyway -- but the bit where Kirk asks why the yet-to-beam-up McCoy was having a "problem" to the yeoman getting off the transporter seems
grossly inappropriate given the accident that just killed 2 people. That was probably taken out for a reason to begin with.
Overall, though -- and maybe I'll change my mind once I finish -- but at this point I'm once again fine with the theatrical version. I don't need the "talking computer" removed, I don't need the various old beeps and blips that "date" the movie as a product of 1979 taken out -- because in the end, it is still is a product of 1979. Warts and all, the theatrical version seems a little more "alive" to me, rougher around the edges, but many of these changes -- subtle as most of them are -- seem arbitrary and don't enhance what was on-screen to begin with.