Going from the scheduled notary appointment to having the first company registered comes out to 17 days in the timeline. 24 days if you add the time for getting the notary appointment. From experience, getting your tax id will take another four weeks or so, depending on your local tax authority. OP is still in that four week window, starting from the point where the paperwork was filed. So about 2 months total of processes you can't speed up.

But notably in OP's case, most of the time is actually spent with the lawyer and tax advisor, no the bureaucracy itself. The more complicated company structure (Ug & Co KG is basically one company (an UG) owning another (a KG), giving a very similar structure to an UG at a slight tax advantage), doesn't help, but it's really not where the majority of the time here is lost (it probably does account for about half of the money though)