OK, the first thing to bear in mind is that there is no substitute for a good carpet on a quality underlay when it comes to sound insulation (in either direction). Everything else is a compromise and the probability is that your floorboards were never intended to be exposed. Carpet would possibly have been laid when your flat was first built.
The other thing to remember is that you can investigate the acoustic qualities of flooring till the cows come home but if the result is that enough sound is transmitted to annoy your neighbors, then you have a problem.
That said, cork does have sound deadening qualities and is used as a layer in some of the higher-end ‘acoustic’ timber flooring. It may even be be better than timber as a flooring material if installed properly. Have a look at these websites HERE and HERE and you could talk to an acoustic engineer.
Judging by what I’ve read, cork boards rather than tiles ‘floating’ on an effective insulation bed will give you a better sound insulation than timber would. But whether that is enough depends very much on the specifics of your building.