From 1958 to 1962 my family lived in a large ranch style house with cork floors throughout most of the home. Even though I was only 11 when we moved from this house I can still recall the way the floors were cared for. On a daily basis my Mom either used her Regina Elektric Broom or a dust mop the keep the floors dust and dirt free. Weekly she wet mopped the floors, what she used I'm not sure, but I recall that she did use Spic and Span on the linoleum floors, so she probably used it on the cork too. The cork floors were regularlly waxed with Treewax liquid. It wasn't a self polishing wax like Aerowax (I'm dating myself), but rather it was like Johnson and Johnson Pride wax for furniture. It was a wax that was milky looking and was applied with an applicator that looked like lambs wool. The Treewax was allowed to dry to a haze and then was buffed to a high sheen with a Shetland electric buffer. The wax would last for several weeks between applications and by simply buffing them again with the Shetland the shine came right back. These floors were very beautiful and somewhat soundproofed the cement slab floors that they were installed over. I recall that my parents were very proud of these cork floors and we didn't know anyone else that had them.
I think that you could safely use Murphys Wood Oil Soap to clean the floors. I'm pretty sure that liquid wax like I discribed above is probably no longer available, but Johnson and Johnson still makes a paste wax and so does Tree Wax, they are just a lot more labor intensive to apply, but the results are worth it and last a long time.
Eddie