I bought UU1+2 and found out that both of games lack Origin/Blue Sky/Looking Glass logos right before the title screen. Sometimes they are just not present, but most times they are glitched, corrupted or being shown for milliseconds.

Is it a bug, or GOG release was specially modified in that way tp avoid possible legal disputes? Is there any way to restore the original title screens? I like how they usually go with the music, and now, without them, introductions seem off. And, after all, I don't want to see graphical glitchfest every time I start these games, and I definitely do not want to see Origin's name being forgotten :(

Looks like this is simply due to fast loading times on modern machines. GOG didn't alter anything with the logo display, but Origin didn't implemented a timer to show it. It seems the logo is only shown to a certain degree of loading the game and then switches to the title screen. Reading data from disks is much faster nowadays, even if you slow down the DOSBox environment the games is running in. When you see the logo, it's probably when you start the game for the first time after rebooting your PC. Then the game is completely loaded from your hard drive which takes the longest time, but is still fast on modern machines. Once you've started it and restart it, the logo is only a blink, because (part of) the data is still cached, so loading is done from cache memory and that's much faster than reading it from disks.
The only thing you can do is slowing DOSBox extremely down by setting cycles to 500 (open \Ultima Underworld 1 and 2\Ultima Underworld 1\dosboxULTIMA1.conf with any text editor and set cycles=500). Then your virtual CPU is so slow, that it compensates the fast loading, but the game is pretty unplayable afterwards, because it's slow as well.

As far as I understand, you are wrong in this case. I have played the original version of the game on Win9x machine and via DOSBox, as well as Pocket PC port, and in all versions logos were precisely syncronised with title screen music.

I just checked this myself by fiddling with the cycles setting in DOSBox while launching the game and it's definitely a speed issue. You have to throttle DOSBox a *lot* to get the logos to stick for any length of time, but at low enough values, it works. (Likewise, the first launch is a hair slower than the second and subsequent due to caching.)

Various DOS games have these issues and they aren't just related to the clock cycles (in addition to fast loading on modern systems, differences in CPU architecture are often responsible for timing-related issues), but very low cycles can mitigate them.

Like Garran confirmed, with GOG's version it's a speed issue. I've tried it a few hours ago myself, with the same result. I even extracted the apparently unaltered DOS version from 1993 included in GOG's image file (game.gog) and put it in my own DOSBox installation, still with the same result - no synchronization to the music. Unfortunately I can't find the original discs at the moment, to compare the hashes or at least the file size of the executables.

Maybe this difference in GOG re-release is somehow connected with disabling of copy protection? Anyways, seems like Garran was right, so I apologise for my denial... I clearly remember original game running smoothly even on powerful 2000s PCs. Unfortunately, I used pirated copy, and I don't want to download it from some nasty site to compare.

AFAIK those games never had copy protection, or at least the original disk versions did not. Even in the main series, the copy protection was of the "what does the manual say on page..." variety.