Hi Sue
I did an almost identical job to this last year and the clients specifically stated that they didn't want any chemicals using on the solid oak floors so I bought a microfibre mop system from Vileda professional which worked a treat. We had to go over the floors twice and they were pretty big and heavily soiled areas with plaster dust etc but using anything else just spread it around and left marks on the floor when it had dried so the vileda mop system and pads was well worth the investment and we now use them on all our other jobs. The clients love the fact that you don't need to use any chemicals too.
Nickie