My cart is similar and currently a much peeling and chipped dark green. I've looked it over with stripping and refinishing the wood in mind carefully. I've done quite a bit of wood refinishing of antiques in the past and the fact is that this is a very daunting job. I decided that it would have to be basically disassembled because every piece of wood on the cart has large bolt assemblies that you couldn't sand nor strip around effectively. In any case, just sanding down the cart wouldn't be the best way to refinish it, you'd need to use stripper to remove all the paint first I think.
I think I've decided to repaint all the wood on mine a color that won't look bad with the dark green metal work, which is a very high quality finish in great shape. Even so I'm facing careful sanding all over every inch of wood to give the new paint something to stick to, partial disassembly, priming, and yeah I'm intimidated! Just the thought of getting the hand sanding right on those wheels....
Good luck whatever you choose, I'm new to driving so can't advise you much but your cart and mare are beautiful!