I put the right batteries in the NIKON F10 and
when I hit the shutter release no lighted up exposure metering
appeared in the NIKON TTL.

If the FM10 works like most manual Nikons, you have to advance the
film counter to "1" before the meter starts working. You should be
able to do this without film in the camera by alternately using the
film advance and pressing the shutter for 3 or 4 cycles. If you can't
then there is probably something wrong with the camera. Try opening
and closing the back to reset the film counter first.

As pointed out above, you may need to have the film advance lever
pulled away from the body to enable the meter, and the button on the
bottom is a film rewind release than you must press before using the
rewind crank to actually rewind the film.

You can probably find an FM10 user's manual with a Yahoo/Google search.