If your lathe is one of the common 1940s makes, Myford, Drummond etc, you can buy a new back plate with the spindle thread and register already machined. Mount it on your own lathe's spindle and turn the back plate to fit your new chuck. This will result in a more accurate job than sending it out to a machine shop. OR as suggested already if the old back plate has enough "meat" left on it, machine that to fit your new chuck.
Of course, best result will follow if you make sure the headstock bearings are well adjusted, ditto carriage and cross slide gib strips etc before machining the new back plate.