What is the correct way to query for traffic_signs in Overpass in an area?

Not sure where you got that query from. I usually either go to http://overpass-turbo.eu/ and put something in the wizard ("traffic sign" works well here) or go to the openstreetmap wiki, look for the thing I am after ("traffic sign") and that gets me here https://wiki.openstreetmap.org/wiki/Key:traffic_sign and there's a link to overpass-turbo.eu that is this:

/*
This query looks for nodes, ways and relations 
with the given key.
Choose your region and hit the Run button above!
*/
[out:json][timeout:25];
// gather results
(
  // query part for: “traffic_sign=*”
  node["traffic_sign"]({{bbox}});
  way["traffic_sign"]({{bbox}});
  relation["traffic_sign"]({{bbox}});
);
// print results
out body;
>;
out skel qt;

This seems to show no traffic signs in my town, although there's some in neighbouring towns so I'm sure its working. Perhaps traffic signs are badly mapped in my town.

Overpass query syntax is a bit of a mystery to me, so I'm not sure what's wrong with yours (except you have plural "traffic_signs") but even with "traffic_sign" I too get no matches. I'm not sure I understand that form of the search query - I usually get the wizards to work for me!

I think your query might be looking for a traffic sign with a name that is literally "*" - I don't see anything in the docs that says "*" is special. But a query with no "=" sign matches all:

["key"]  /* filter objects tagged with this key and any value */

From https://wiki.openstreetmap.org/wiki/Overpass_API/Language_Guide