Well, finally used two days of a seven day free trial of CBS to binge Disco and see what I thought.
After watching the show, here's how it resolves in my head if I try to force it to make any sense at all:
TOS is it's own timeline- the 'quantum signature' of that universe would also be different than Disco's. There's that problem solved.
I guess that the prime timeline splits, forks, or whatever during the events of First Contact. Zephram Cochrane and his people are exposed to 24th Century tech and knowledge of the future, and Borg debris from the 24th Century is left on 21st Century Earth. The Enterprise timeline springs from that, and the Disco timeline is Enterprise's 23rd Century.
The Klingon disparity cannot be resolved, except to say that Discovery's timeline is yet again an alternate of the one I described above: like in that episode of TNG (Parallels) when Worf is hopping universes, and some of them are so nearly identical that the differences are like having a different painting on the wall. So that would be to say that Discovery doesn't spring from the ENT timeline we saw, but another parallel very similar.
Discovery's uniforms, production values, and starship designs all look like a natural outgrowth of what we saw in Enterprise. They don't reconcile with TOS, and after 50 years I wouldn't really expect them to anyway. Enterprise's tech/ productions values didn't with reconcile with TOS. It also looks to me like CBS has attempted to make Disco's production values fall in line with the JJ-verse movies. An understandable move on their part but destined to rouse fan-ire if they are going to insist on calling it the prime timeline.
What of the show itself? It's dreck. The writing, dialogue, characterizations- everything is a hot mess. Sarek and Amanda appear younger than their daughter- I guess that's possible with Sarek, but not Amanda. Having them in the story at all- totally unnecessary and gives it all 'small universe' syndrome like SW suffers from. The capper for me was when the Sarek agreed to putting the mirror-universe emperor in command of Discovery to commit genocide against the Klingon homeworld. Getting to that point was bad enough- that little tidbit was the point where I mentally struck my colors.
The fx were gorgeous- I'll give it kudos on that one point.