That is a National SemiConductor module and replacements are pretty easy to find.
Does the watch display anything at all? From what I can see on the photo's the module looks clean and the battery contacts are still intact. Try gently cleaning all the contact points with a fibre tipped pen and retrying with fresh batteries.
That particular module is prone to quartz crystal failure, replacing the QC is easy (even I have done it) and a new QC costs only a few pence, a single digit on the display is usually a good indicator that the QC has gone bad.
If all else fails and you need a new module contact Ed Cantarella at
retroleds@aol.com I would be amazed if Ed does not have a spare NSC module!