To date, none of the reference book info I’ve come across shows Lyon & Healy owning Regal - although it’s certainly possible at some point they did (but must say, I do trust books more than the internet!).
Regal did have its beginnings in Indiana, while Lyon & Healy was formed in Chicago by the Oliver Ditson Co. of Boston (for whom Martin would build the first dreadnought guitar). This new enterprise was named for the Ditson associates who ran it - George Washburn Lyon & Patrick Healy. By 1880, Lyon & Healy was independent & sold instruments under the Washburn name - and they built a large factory to produce a rather extensive line & variety of instruments. The Washburn brand flourished until the 1920s, but then sales dropped off, with Lyon & Healy getting out of the guitar business in 1928. They sold the Washburn name to the Tonk Brothers, who eventually hired Regal to build Washburns. Regal was already now established in Chicago, primarily building mandolins. They took over the Washburn factory for Tonk Bros, and simultaneously began an expanded line of Regal instruments. Regal’s heyday was in the ‘30s. They’re probably best known for making Dobros for the Dopyera Bros from 1933 to 1942. While operations continued into the ‘40s, they never totally regained their footing after WWII.
Indeed, establishing a build date for this guitar should be helpful in figuring out which company/factory built it, so let us know if you find out more!