I could rant about the annoying nature of this problem happening in the first place but that's been done enough. After lots of scrolling through the internet for answers I stumbled across a solution to my similar problem.
I too was very annoyed to discover when i wasn't able to connect to the internet that I was unable to access most of my files. Ridiculous that this could happent without a VERY clear warning .... but back to the solution.
This might not work for other people but when I was back online I was just looking at the One Drive folders... Pictures, Work etc and when i right clicked on a folder it had the option of "Always keep on this device" There was a transfer moment that didn't take too long, suggesting the files were always somewhere on my computer to begin with, and now there is a copy back on my hard drive.
Hope this fixes the problem for at least some people. If not, can't help you any more, just blind luck I found this out at all.