What Is The Correct Way To Recombine A Split Serial Signal?
I need to replace the left hand side of the schematic with an Arduino
  or similar


This statement does not make any sense. The left side of the schematics is just an interface, you do not "replace" it, you connect "Arduino or similar" board to it. After making sure that voltage levels are compatible, of course. But that is a simple task of choosing correct optocouplers and adding resistor to TDO line.

The only thing that might require a bit of tweaking is the signal polarity. But even that can be sometimes resolved in software since many MCUs have "inverted" bits in UART control registers.