Ok, I'll probably catch some flak for this and I'll preface it by saying I have no experience in any actual film production...and I Have nothing but respect for all fan productions. Regardless of the quality in the final output, I give them credit for the work they have done and what they have achieved.
That being said, there is a problem I have noticed in many fan productions. For the most part, many seem to intent on trying and recreate some aspect of one various series. Regardless of whether it's a crew on a ship or a crew on a station, there seems to be a strict adherence to the episode/season format. This ultimately hurts many fan productions I think.
Instead of trying to create several episodes over a long period of time, why not try to make a movie, or a mini-series with one overall tightly focused story. Regardless of the story you tell, I think this heightened focus on a smaller project can improve production because you're giving everyone involved a very clear goal to work towards.
So rather than trying to recreate TNG in the Abrams verse, figure out a story you want to tell, then figure out a way to film it. Once that story is complete, tell another one.
Because let's be realistic and yes, I realize that this goes without saying, but I'll say it anyway. We're not hollywood. We don't have their resources or their budget. Even the most well financed fan productions still have trouble completing episodes in a timely manner. And from what I've seen, most fan productions stick with that episode format as if it's the only thing they are allowed to do. Don't venture too far, or you might do something original.
So why not take some chances. I know that if I came across two fan productions, one that was yet another crew aboard a starship series and another that was telling a unique story about some small personal aspect of the Star Trek universe, I would be much more interested in the latter. Taking the risk and doing something that hasn't been done often or at all, goes a long way in my book. Perhaps I'm in the minority, but I at least hope you give this some consideration.
Don't worry about pilots and season structures. Figure out a story you want to tell and focus on telling it as well as possible. If the story is good, the audience will forgive other things like poor acting or bad CGI.
As for story ideas, if you have Picard already cast, than obviously make him the center of the production and I would suggest making it a personal piece about him.
I think it would be interesting to start out doing some things set on Earth. We have earth all around us, so why not take advantage of it before you have to get into space and CGI. Ground things in reality with small everyday tasks. One of the things I love about the Abrams films, is that there are simple everyday occurrences...then you look at the bigger picture and realize they're in the future when a shuttle craft flies by.
We see Kirk in a bar, or riding a moto cycle, or driving a car, and in the previous film you see parents watching over their sick child. Those are all things you can point to and say, those are things I can relate to. I've done those things or I know someone how has. It makes things real and adds a level of authenticity to the story without having to exert a great deal of effort.
So, show Picard at home with his brother and have him chide Picard for missing his wedding. Mention Marie is pregnant or has just had Rene...just brainstorming here without much thought about the specific timing of when events in his life would actually occur.
You mentioned the Stargazer and we know Picard faced a court martial when the ship was destroyed, so you could show that as well.
It could be interesting showing a character coming to grips with having lost his first command during a court martial....featuring flashbacks to the action that occurred aboard the ship as Picard gives his testimony. He saved some of the crew, but ultimately lost the ship. Bring up whats her name that acted as judge during Data's trial against Commander Maddex...
Jack Crusher might also be around during this time as well.
Few if any actual crew were actually mentioned by name so you could go with as many new characters as you like.
If Picard was the commanding officer of the Stargazer, then he might feel obligated to visit the families of some of the crew.
If your actor is up to the task, this could be a really great character piece.
The ending....Picard is acquitted of course, but they still don't know who the aliens actually are who destroyed the Stargazer. If you recall, they didn't actually make first contact with the Ferengi until later in TNG proper. For some reason which isn't clear, Picard is given a new ship to hunt these aliens down and see just how dangerous they really are.
I would end it here and then make the second story about Picard wrestling with his orders to investigate the Ferengi threat...it's clear Starfleet expects some sort of retribution, but Picard is determined to find a peaceful alternative.
With some careful work, the story can serve as both the set up to the second part when the Enterprise is finally launched and as a standalone character piece for Picard. If nothing else, you'll go into that second episode with a well developed established character and one production already under your belt. For my money, you'll also have the bonus of doing something that is unique, making a fan production that isn't just a recreation/spinoff or continuation of the series that have come before.
Again I don't intend to any disrepute towards any of the fan productions currently taking place. The bottom line is that were I in your position, I would focus my efforts on something other than trying to recreate the ship/station/crew episodic format. To each their own though because as I understand it, fan productions are all passion projects so there's really no right or wrong.
Anyway, The big question of course is how the movies will effect the TNG era. Personally, I think many things have been accelerated and Starfleet's apparent rush towards pursuing advanced technology has not only put them into contact with a number of species they might not have met until later, but also created a sort of cold war arms race throughout much of the quadrant.
Some sort of general timeline of major events might be required, just so everyone involved has a general idea of where things stand.