Help identifying potentiometers or switches on an old oscilloscope
One of the largest component suppliers in the UK /World calls them flat shafts.  However, these are uncommon today and I can only find a very few on the RS Components web site.  They mostly seem to be fully round.  They typically come in 6.00 and 6.35 mm diameters so aim for the same size, but I guess 0.35 mm is neither here nor there for a  knob control.  6.35 is closest to 1/4 inch for your old dials.

Don't let that bother you.  Get some of these:-



They're designed for putting flats on pot shafts.  I don't care what sort of ultra sophisticated engineer you think you are, if you haven't got any files then in my mind your not a real engineer :-)

EDIT.

Stuff it.  Following your edit with the additional photos, Trevor's right below.  You'll have a very hard time finding a fine /coarse pot these days.  But that's irrelevant.  If you're trashing the insides and going digital, then the control panel doesn't have to have full original functionality.  You'd find it difficult to replicate that functionality anyway.  Just use normal potentiometers instead. You could of course rename the dials and have a dedicated fine and dedicated coarse knob instead sacrificing some others.