I think I just figured out what happened. 
So, I decided to take a break from my usual save, and pulled up my nephew’s. I went to the first Interface, and when it came time to make the choice, it suddenly hit me… I “refused” when I originally went to it in my regular save!
Looking at it now, I can’t even believe that’s an option. 
Apologies if this is late but I came here to report this same issue. I have also sent a report to HG after spending an inordinate amount of time troubleshooting.
@Topher can you confirm if you’ve received the “blueprint added” message similar to what Alex reported?
Btw: if you take the coordinates of the current system you’re in, convert them to hex and replace the last 4 values with 007A you will end up in a system with an Atlas Station. Just warp through and set a way point in your discoveries.
@RyanM88 I never received the blueprints like Alex did. I am pretty sure my issue stemmed from refusing to submit to the first Atlas Interface when I originally started the game.
I will try that. I had one set, but it just seemed like I was never going to get there. 
So in a different save file, I started the AP questline, thinking I could make a spare as I went along, leave it in a storage unit, and visit the base in my original save to pick it up. That isn’t working either; the base isn’t showing up despite having uploaded it numerous times. I’ve tried with MP turned on and off in both saves.
In fact, the planet is quite a bit different.
Bummer. I think I’m in the exact same boat (refused to submit the first time). As a side point, if the blueprint you receive after completing The Purge (Remembrance) is any indication, the Atlas Path quest line should be resumable. Otherwise, there would be no point since it requires all of the Atlas Seeds anyway.
Good luck, though. I’m arranging to have the Atlas Seeds sent to me and will report back what I find.
Edit: Source for the Atlas Station coordinates - hover over the glyphs for the tool tip.
You will not be able to see bases from your own account if using a different save. You will only see it in the save it is created with. It requires a different account to be able to see the base as an upload.
So I did more troubleshooting. Turns out you should still be able to continue the Atlas Path if you refused the Atlas during your first visit to an Atlas Station (according to another NMS player).
In my case, what I did is refuse the Atlas during this first visit, but chose to reset the simulation and traveled to Eissentam. At the end of the Purge, I received a quest line to construct the Remembrance. Not knowing that it requires Atlas Seeds, I attempted to construct it which caused the state of the quest to be marked completed.
Going purely based on my own troubleshooting, I suspect that what’s happened is that the game essentially thinks I’m done with the Atlas Path quest (since I ‘constructed’ the Remembrance). Essentially, I can’t go back and obtain the Atlas Seeds myself because a higher state in this particular mission has been attained. I think this is a quest design flaw, and not a bug. It’s the best hypothesis I have, though I could be wrong.
Would anyone else be able to list out what they chose for both the Atlas Path quest line, as well as the Purge/Crafting the Remembrance?
edit: I’ve e-mailed HG again because at this point, I’m confused. If the intended result of refusing the Atlas on first visit to an Atlas Station is to stop players from obtaining Atlas Seeds, then why do you receive the Remembrance anyway? This is just confusing and should probably be addressed in some form.
So I created a new account and ran through the Atlas quest right up to the seed needed to complete Remembrance. I dropped it in storage, copied the planet address, and loaded up my original. I portaled to the base planet, and found the marker. As I was heading over, the thought hit me; wait, will I see the base owners items or my own? And the answer is, my own. Damnit!
Hehehehe, yup, that’s how it works!
If you have access to multiplayer, you could transfer the item(s) to someone you trust then have them transfer it back.
Ok. So yesterday I crossed from Euclid into Hilbert Dimension. I did get to choose if I wanted to birth a star or continue without doing anything. I got a new Star Seed and I chose to birth a star. It also reopened the Atlas Path for me.
I can still travel freely back to my Euclid base and I had no issues calling in my Freighter. I also worked the weekend mission from another galaxy and MP with a guy from PS4 though I failed to ask what galaxy he was in…
Anyway, maybe that info will help someone