What specific primer can you get? Also - what cases and what powder? The thing that immediately comes to mind about NATO primers is the cup thickness / hardness. They are designed to prevent slamfires in various semi-autos and meet mil-spec requirements meaning they have a bit more heft to them compared to standard LRM primers. (They are also considered ‘magnum’ to my understanding.)
Personally - I’d be looking for a different primer, but also know the times we are in (maybe you could do some trading?). Would NATO primers be ideal for the 500 in a revolver? Not really. Could they work? If your revolver can reliably ignite them - and you use published data - I don’t see why not. Reminder: start low and work up.
The 500S&W was originally designed to be used with a large pistol primer, but that design was changed in 2003. S&W and Cor-bon changed the design to now include the use of a large rifle primer. (
https://www.starlinebrass.com/500-sw-mag-brass)
Unless the OP managed to find nearly 20 year old+ brass, he should be using Large Rifle or Large Rifle Magnum primers (based on published load data for his powder/bullet combo).
OP: obviously - check your brass to ensure you’re using the correct primer size.