Here’s your lavatory faucet.
https://www.houseofantiquehardware....ad-bathroom-faucet-with-5-spoke-cross-handles
I would assume by looking at them that the tub would be the same. Maybe not.,
To be honest I believe these faucets take a generic stem that can be purchased and it fit several different makers of valves.
Remove some of the decorative trim and I might could tell you more. But ultimately a plumber needs to remove the inside parts and take pictures, measure with calipers. Then put it back together if he doesn’t have the right parts. Then order parts.
In this situation it’s more economical than replacements.,
The lavatory with the missing or broke spout may be at the end of its life. Does it screw in ?
If I’m ever in Tennesee the beer and burgers are on you.....