The Champagne Caves, also known as Oil Mills West and Finnis Hill Caves, are of 19th century date and are reputed to have been built by Napoleonic prisoners of war, possibly taking advantage of the earlier lime workings suggested by the road name (lime burning took place to create, ultimately, lime for use in building mortar and plaster).
They comprise five mainly brick-lined tunnels running into the cliff face in two parallel groups (of two and three tunnels apiece) which may reflect different building stages. An upper floor may have been present at some time in the much taller east end caves (the word caves is used for convenience however these are man-made structures) and current access is via upper floor openings
This appears to be the site described below in a fascinating website on Dover breweries and pubs: “Caves situated in the limestone cliffs of Finnis Hill”