It's probably because you're looking at your research from the wrong base. You have to select the base where your scientists/lab actually exists to use them. For example if you have your scientists in base 1, if you look at your research page from base 2, the scientists will show up as 0 (10). It's a pretty annoying mechanic of the game where scientists and mechanics don't just share one global pool.

I did some more checks and the 10 scientists and 5 engineers disappear from use but are still employed. Hiring more doesn't fix the issue. But I can use the new hires. I can't fire the 10 or 5 either.
I can progress and will just accept it. I'll just have a small handicap going forward.

I'm running into the same issue. A third into the game I have no more available researchers. Sometimes the research doesn't go through and it once happened with alien biology so it broke the game.