How long are TV shows typically awaiting airing after completion, Star Trek: Picard in particular?

I'm not sure there is such a thing as a 'typical' lead time. It's very much dependent on episode count per year and budget.

I've never worked on a sci-fi big budget show like the Star Trek franchise, so I can't be specific on that one, sorry.

I've worked on 8- or 10-episode flagship shows where principal photography was May - October, with 1st episode xmas then the main season starting January/February. Shows such as Call the Midwife, Mr Selfridge and Downton Abbey would fall into this category.
To break that mould, Silent Witness shoots a 10-month schedule for 10 episodes, but they run something like 2 months on, 1 month off, which is an unusual setup.

Mammoth budget stuff like The Crown is maybe a year from start of principal photography to air date - but that being Netflix is all released in a single day, which means the important lead time is from the end of photography. For instance, season 3 aired November, by which time we were well into season 4 filming.

Conversely, the large budget show "24" were filming 'dangerously close' to air date - sacrificing location efficiency for sequential scene setups. We would go back to a location for every episode, no matter how costly it was compared to shooting all one location's requirements in one visit, because at some points they were only two weeks ahead of air-date.

I've worked 24-episode shows and 200-episode soaps where the lead time 'standard' is closer to two months. The show I'm working at the moment feels something similar to this, 10 episodes; photography ends at the end of February, air-date is May. [It's going to be on Hulu and I don't know whether they do the Netflix all-at-once or the Amazon one-a-week structure].

One very long lead time movie I know of was Dumbo. That was shot across Summer 2017 but not released until late 2019.

After your edit -
Some streaming services release all at once [Netflix model] some release one a week [Amazon model] Some terrestrial services do both - broadcast to air one a week, but drop the whole lot to streaming on day one [BBC do this sometimes with iPlayer].
That has to be factored in before production starts. It still doesn't mean there is a 'typical' lead-time.

Also bear in mind these days one director rarely directs the entire series. They get a block each, so can go off to edit their block as soon as it's finished.