Drill holes in the back case for all the things that need to be mounted there. The power jack can probably use the hole for the original power wires. The charge indicator LED needs a hole just over 5 mm that it can be glued into. The battery switch needs a rectangular hole about ½"×¾", so drill a ½" hole and carve it to the right size and shape with a rotary tool and/or a file. The antenna jack needs a ¼" hole.

Mount all the back panel things to the back panel and add wires and connectors. 6 inch wires are plenty. Be careful about the polarity of the power jack and charge indicator LED.

Put the battery in the 3D-printed battery holder, and mount it in the back case. Add the right type of connector to the battery wires, being careful of polarity.

Populate the circuit board by whatever method you're most comfortable with. Note that the surface-mount dual transistor packs have two-fold rotational symmetry, so they can be installed backwards without hurting anything.

Mount the circuit board in the traffic light. Connect the antenna, power jack, battery switch, battery, and charge LED to the circuit board.

Connect a USB-UART adapter to P3 (sorry about the weird pinout, it matches some other weird things I have). Flash the ESP-07 module with MicroPython. If you want to be able to display PNGs, add png.py from my MicroPython PNG library (and itertools from micropython-lib) to the esp8266/modules directory before compiling MicroPython.

Wait patiently for me to polish the software a bit before putting it in a git repository.