In the 1800s here (US) it was assumed that any decent farmer should be able to make a new wagon wheel, or rebuild a broken one, with an axe and a scrap or two of wood - as long as the two or three iron parts (mostly rings) could be hammered straight enough. If you have iron spokes, it likely takes a little more acquired skill, and you might have to build yourself a back yard forge.
In the UK, for the time suggested, there was probably more common use of "factory made" wheels less amenable to home repair.
In either area, there was sufficient variation in methods that "picnic carriage" doesn't say much (to me) about the specifics of the wheels you want to fix/replace.
It might be more likely to elicit useful suggestions if you could provide more specific description of the original wheels - size and construction - for which you need replacements(?).