"... Isn't there usually a plywood layer below the hardwood floors?...">No ... that is the way older homes floors were installed before engineered panel technology we enjoy today came to market, and considerably after they entered the market. Some home builders still will install thicker wood flooring directly over joists. I think its improper now however and a engineered panel subfloor is now the norm.
"... We are thinking of installing cork, could the existing wood floors now act as the subfloor? Will we need to install a layer of plywood on top of the wood floors to install cork ...">In general terms, The least I would in your situation do is install a layer of 3/4" engineered panel over that plus a layer of flooring grade underlayment if instlling a glued-down cork product.

... and if existing wood plank is in bad shape, demo the planks and install two layers of engineered panel over the joists.

In more exact terms, you need to follow the installation specifications from the manufacturer of the flooring you choose. Specifically the prerequsites for subfloor type, construction, and condition.

Just ensure yours falls within their specs.

Cork comes in so many types and varieties, from resilient sheet and tile that need glued, to layered click-in planks and tiles, so ensure you adhere to the manufacturer specs.

If unsure, post the specific product and we can research it for you.