How have you got on with this chair ?
The first thing that I would do is take the chair to pieces , by undoing all the bolts , and then see it the the different parts keep their shape .
If they do , I would lightly sand , and to nurture the wood , oil ( with a plain wood oil , not a mixed one) the rockers , sides/arms etc , but not the seat or the back , at this stage .
Then I would do as those above have suggested with the seat and back , cleaning it down , replacing the cane , plain oiling and allow it to soak in deeply , and to dry on the surface.
When it is , I would then apply a coat or two of an oil mixture , allow to fully dry , and reassemble with new bolts nuts etc.