The best solution is to disassemble all, like that you'll can clean well the keys.
When I made the retr0bright on my wurlitzer keys I have protected the wood parts with a tape. At the same time I put the solution on a extra wood part to see if it could be damaged, and it made nothing to the wood part.
So I'd like to say that the wood will not be damaged, but it depends on the stain, varnish...