Early axle grease was tallow, made from animal fat rendered from beef or mutton. A less desirable alternative, sometimes used in a pinch, was rendered bacon fat, or lard. Save your rendered fat.
I wonder if HPL makes a Super-buckboard, euro version. Schaeffer makes grease that's suitable for buggy and wagon axles. There are at least a few wheel and wagon shops that carry it.
Typically, about every 40 miles, although some wagon makers (and wagon masters) recommended 3x per week. Of course, that depended upon the conditions in which the wagons were used.
Your wife is a keeper. First thing I’d do is treat the wood and metal surfaces with suitable preservatives. That is a real historical artifact and I’ve seen way too many of them rotten and falling apart.