Horse Drawn Wagons
This is kinda over-detailed and probably would never feature in a campaign, but it's just some thoughts on why you won't see old-style wooden construction wagons in the Twilight World.
Wooden wagons are insanely complex and require wood crafting skills that simply don't exist on a large scale and take far too long to relearn or teach in the time period. During a national emergency you don't take a huge amount of people out of the workforce and teach them an ancient, almost forgotten-at-the-time massive array of skills. Many years ago as a re-enactor I wanted to make a comparatively crude wagon wheel and I was amazed at the amount of knowledge and equipment it took to make a proper wooden wheel let alone a functioning axle.
But that's not important because welded mild steel is lighter, in some ways more flexible and also more rigid where needed and well understood. Light rubber tyred-wheels are widespread and motorcycle and off-road vehicles make these wheels and axles. There's only a short window where criteria and perhaps even standard plans can be created but I think this would happen.
What would a "developed" version look like?
First off, unless it's cargo-bearing it might not be a rigid frame but rather two articulated units for better manoeuvrability such as some German examples from The Second World War.
Simple box steel frames support a lightweight cargo bed if fitted, probably wooden but thin steel isn't that different in weight. Drop down sides for some cargo versions might be fitted and users could experience the joy of having their fingers mashed when the other person suddenly drops one.
A standard width would allow bows and covers from existing cargo vehicles for light weight weather protection. A lightweight small generator platform under the cargo bed allows for radios and a standard aerial mount would go somewhere, probably forward and just behind the front passenger, however this puts it at risk in fording procedures and a space between operators and payload area might be better. For local protection a small frame between the driver and assistant would hold individual weapons and under their vehicle seats, probably the same all-weather seats from some light vehicle, would be space for ammunition and personal storage. If a battery is fitted (and I can't see why not) there'd probably be convoy marker lights for low visibility mounted on the extremities of the vehicle.
There has to be space for pioneer tools and also to store the horse tack and harness and this usually goes between the payload and operators as well, shifting the power to one side in a metal box.
Obligatory German versions:
