Very sorry to see another floor ruined with "oil soap". This stuff should be banned...but I won't go there. The most effective removal of old cleaning products = "screening" = machine removal of the cleaning products. This can be done by a professional = the same machine used to sand the floor. The difference is they use a screening pad instead of a sanding pad. The screening will remove as much as 99% of the cleaning product. Sadly, you may never remove all of it...which means it probably can never be refinished until you sand it all the way down to barewood (sanding will take the floor down another 1-3mm).

You could look into a chemical floor stripper that is used in conjunction with the screening. But again, there is really no way to know how much you removed until you sand it down to wood. You will not be able to do a "buff and coat" or reapplication. Your only option will be to sand and refinish...and even then, the finish may not grab.

You say this is your floor with no subfloor. I have the same thing in our old house here in Texas. Do you have a lot of movement as you walk on your floors? We do and I have been concerned there is something wrong but was told it is common. I have it throughout our house. It is pretty but can be cold and noisy. Would love to have a subfloor with this flooring on top but would be too costly to do. Good luck with everything