Ok, after this problem nearly drove me mad it seems I found the cause, but it sounds a bit weird:
The *stuck* seems to happen mostly because the game sometimes either misses ships that have still moves left or it wrongly expects me to move ships or fleets that are parked on guard or sentry orders.
So I get a greyed out "wait" on the turn button, when instead there should be either an "idle ship" msg (when I need to move still) or it should just progress normally (when it happens to ships on guard/sentry).
Strangely enough when being stuck on that greyed out "wait" the game seemed to demand higher CPU/gfx performance, as I noted the fans in my machine turn higher.
I found out by accident that when "stuck" searching through my ships/fleets and moving those that still can the game suddenly started the next turn for the AI normally. Also I started to move ships on guard/sentry - that helped too in some cases. So I can progress to the next turn that way an the fans go back to normal.
It still is a problem which should not happen, because it can get a royal pain when I basically have to move everything around manually, but at least I don't get stuck anymore then.