Its quite a nice looking watch.
The movement number indicates early 1960's but the case number doesn't seem to match. The cases at that time started with "36".
However, the case design is similar to the Chronometre Royal of the late 1950's-early 1960's (see The Quater Millenium Antiquorum catalogue...#13 and #14.
You can check the movement type (not the S/N) which should be engraved on the movement somewhere. Can you upload a photo of the movement itself?
Having said all that, the appearance does look like a JLC but there is a similarity amongst watch brands in dial and hand design in that period (as well as earlier periods) so thats not conclusive at all. Unfortunately my JLC references are not extensive. I have a JLC watch from 1950's with a similar appearance.
Perhaps it is a se-called "frankenwatch" were the parts are all genuine VC but come from different eras.
VC in Geneva may have to be the final arbiter.