I’m largely posting this because I’ve had to do it twice recently and forgot how to do it both times.
The command to convert a whole folder of images from .bmp to .jpg (on Linux, obviously) is:
mogrify -format jpg *.bmp
It’s really that simple, and is actually a lot quicker than I thought it would be. You need ImageMagick installed, but most distros will have that by default anyway.