Why do you want a bi-folding door? They're quite complex to produce even for the seasoned professional, with seasonal humidity changes it's not uncommon to see all the doors shrink and expand 15mm+ over the span of all the doors which can make sealing them and making them work properly a nightmare, even in hardwood. I've made quite a few from timber now, my consensus is if you're not going to make it from Accoya it's not worth bothering with, get an aluminum one and forget about it. If you've got very little woodworking skill and knowledge on how wood behaves I would've even consider attempting to make a bi-folding door. Not to mention the gear is priced ridiculously. Depending on the size of the opening you're far better off with a pair of french doors, even with a side-light/panel each side of the doors to block up the opening, they're far more reliable, maintainable and even more in-expensive comparatively. You'd be looking perhaps £150 in hardware all included for a french door and well over £800 for a decent bi-folding gear before looking into handles and such...
I wouldn't use WR Cedar, it's great for external constructions like garden furniture and cladding where you'll leave it and never put a finish on it again, but I wouldn't use it on external joinery simply because it's very expensive for what it is. Accoya is a better alternative for pretty much the same money, if you wanted cheaper you could go with pretty much any of the usual hardwoods like Sapele, Iroko or Utile which will be far cheaper than WR Cedar.