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 and plugin 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, but I'm not sure if that ever actually happens.
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: Try manually emptying your cache. In Chrome, go to 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.
I apologize that it's so confusing. I wish it wasn't. But I'm not sure how to do better. The root of the problem is that the Internet is a complicated place. There is no one thing called "the cache" and there is no one way to say "give me the latest version of X". Different systems have different ideas about all of those things. This article describes a little of the complexity, if you're curious.
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!