If there is no grub screw in the narrow waisted part of the handle, it is all one unit and can be removed by taking out the screws around the edge of the flange. They look to be somewhat damaged by previous work so I would advise you to replace the screws as a matter of course
That will just be a single square metal rod running through the door from side to side. One side will be in the round handle on one side and the same on the other. Just unscrew the 4 screws on the round handle then pull the handle back off - you'll see a square metal rod. Rinse and repeat for the other side, then you can just get any other bog standard handle and stick it on each side. The square metal bar will be a standard size.