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One thing that wouldn’t work is fitting the wrong size, so you would have a tough time mixing, for example, 9.00x16 with 10.50x16s on the same wagon.
Tread pattern is less critical, given that they weren’t racing the things so precise balance and grip were secondary requirements.
I had already assumed the wheels, and correct hub and their positions, would be the same size/type to any one single vehicle, along with a variety of tire...but stupid me forgot to mention that originally.

I guess I'm going for the mixing of tires from different manufacturers!
Robin...again thanx for your input once again. My quest for mixing tire types, to include different tread pattern as well, is the goal. As far as the Germans goes when mixing up tires, yep, they used what was available...no matter the country of origin. I think one example of that is the Famo showing an entirely non standard tire in a few photos.