Basketball GM stores all game data on your computer, in your browser profile. This means that you can't play one league on multiple devices unless you export it (from the Tools menu) and then create a new league with that file. So first, make sure you're using the same browser on the same computer.
If you are using the same browser on the same computer and your leagues are missing, the game data has probably been deleted. This can happen in places like schools and libraries that set browsers to automatically delete everything when they are closed. It also happens if you manually delete your browser data. For example, in Chrome, if you go to More tools > Clear browsing data... > Cookies and other site data, that will delete all your Basketball GM data. This is true even if you tell it to only delete data from today - if you played Basketball GM at all today, it will completely delete all your leagues.
Browsers may also delete data if disk space is running low. As far as I know, this only actually happens on mobile.
In cases where the data has been deleted from your browser profile, the only way to get it back is if you have a backup. Sorry :(
Currently there's no easy way to do this, but it is possible.
You can export a league (Tools > Export League), copy the file to your other computer, and then create a new league using that file.
Easy answer: You'll get it automatically, eventually.
Medium answer: Close all tabs with Basketball GM open and then open Basketball GM again. Or restart your browser. If this doesn't work the first time, try again. Pressing the "reload" button will probably not help.
Hard answer #1: Try manually emptying your cache. In Chrome, go to More tools > Clear browsing data. Make sure you select "Cached images and files" and do not select "Cookies and other site data" because "other site data" is where Basketball GM stores all your leagues. Then try closing all open tabs with Basketball GM and opening it again, like above.
Hard answer #2: Open Basketball GM in a single tab in your web browser. Open your browser dev tools (ctrl + shift + i) and then do a hard refresh (ctrl + shift + r).
Not in the near future, for several reasons:
No. I'm not convinced that playing time really matters for player development.
It's a chicken and egg problem. Do good young players get more playing time because they're already good? Or do young players become good if they get more playing time? It's obvious that the first scenario happens. But is it obvious that the second scenario happens? I haven't seen any convincing evidence that it does, and there is some evidence that it doesn't.
I know that's not really a fair argument, because it could be that it's a real effect that is just hard to see in the stats we have available. But if it's hard to see in the stats we have available, does it matter for a game like Basketball GM? Because if it's so difficult to notice in reality, it'd be equally difficult to notice in Basketball GM.
I can make a counterargument, though. Basketball GM is a video game. Its primary purpose is to be fun, not to be realistic. So maybe it'd be fun to give unrealistically large bonuses to young players who get a lot of playing time. I don't know. It could be fun. But I also very much like to keep things simple, so I err on the side of simplicity. And doing nothing is very simple.
I also am aware that I could be wrong. Maybe there is a convincing, data-driven argument that playing time leads to quantifiable improvement. If so, please let me know!