From this single image, it's hard to tell exactly how the switch and lamp socket are conjoined (if at all) and if the socket is [commonly] attached to the porcelain fixture along with the switch through that single hole thru the use of a custom mount attaching to the base of the socket using standard ⅛" pipe thread, or not.
If the original lamp utilized a "custom" combination lamp socket/switch, you may need to custom-make a bracket to secure a [modern day] lamp socket that then is secured to the fixture using the mounting shaft of a new switch, thus securing the entire assembly to the fixture. Good luck.