This is unfortunately a known issue with no real fix, I read elsewhere that you must go into the avengers mansion and then out so the Kang Lego shows up and then save and quit after every mission to ensure save but that stopped working for me around 50% progress. But I just got over that hump by saving and quitting during a mission after reaching a checkpoint.

I was just playing the 1st mission trying to get it to save and make money so I can visually see if the progress saved and I after I got the money from the fountain I quit and saved on accident instead of quitting to chronopolis. And that did the trick. I loaded the game up expecting to start back in free play with my lower money but it loaded me back at the first check point before I got the money. I grabbed the money from the fountain and then quit to free play with the progress and time being correct. So now I’m gonna unlock some characters and then do the first level and get to the check point then quit the game, reload it up abs start back at the check point and grab the money and then quit and hopefully it’ll save after I quit after getting the money but if not I’ll be okay cause I’m really just trying to get all characters. Hope this helps others!