The wood is very dirty and its been hard to tell, we were actually just looking at it and thinking it may be chestnut. It's a very hard wood.

Walnut with burled walnut veneers on the two top raised curves . It's hand caned and will cost a small fortune to have the cane redone :(

Here's a close up of the carvings

You can learn caning. It's a tedious PITA, but not difficult.

up close it still appears to be Walnut , but not veneered on the top curves , solid through out .
you can learn to do anything , hand caning is not something ya that just pick up .... and doing a curved back and seat is even harder to do then a square .

I think it`s 18th century, the nails are a clumsy way to repair a useful piece of furniture, It would all have been held together by pegs/dowel. It`s walnut, as stocky says, and should be restored by an expert. A wonderful chair.

It looks Biedermeier to me ... except for the turned legs,

It's worth taking to a furniture expert.

It is held together with pegs/dowels! Fantastic thank you for the information greatly greatly appreciated.

We have a second chair that is constructed similarly, they came from the same place. This one is walnut as well.

Is there anything that you can tell from this chair that you could not from the first one?

Your second chair is a mid-Victorian "sewing" rocker.