I contacted Longines Switzerland about this watch. This is what they had to say:
Dear Mr. Sanders,
Thank you for your e-mail as well as for your interest in LONGINES watches.
We have pleasure in giving you below the information we found in our old, hand-written production register regarding your LONGINES watch.
Serial number 6'857'828 is a wrist watch in 14k gold reference 5603
It is been invoiced on 6th July 1945 to Messrs. Wittnauer, who were our agents in the U.S.A.for many years.
It is equipped with a movement calibre 23 M
Estimation: Swiss francs 1'200 (in good condition). It is very difficult for us to make an appraisal from afar without having the watch in hand.
Hope this information is convenient to you, and always at your disposal, I remain with my best regards.