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To prevent them from showing through the floor, there's the chance that either the gap between countersunk screw head and floor will show once it's been walked on a bit (or glued down etc.) or that the screw head will be slightly proud (in which case filling will show it up). I guess movement or swelling would be a secondary consideration, especially since it's always worth figuring a flood of some small type every 10 years of occupation, accidents do happen!
Sounds like you have understood the process.
Remove existing tiles
Remove damaged sub floor (probably ply, possibly some kind of tongue and groove OSB).
Fit 18mm ply subfloor fixed to joists at 200mm centres with countersunk screws
Screw heads filled and sanded
Lay cork tiles on adhesive recommended by cork tile manufacturer.
You have a choice whether to remove the skirting first.
I wouldn't under estimate the screwing down and filling of screw heads. On some floors this whole process adds a days labour. I use a 2 part filler because it worries me that a powder filler can break up but I do feel sorry for anyone who has to undo it. It's pretty normal for a thin ply layer to be attached to an exiting subfloor if it's in reasonable condition. With any glued cork or vinyl tile floor the sub floor must be very smooth and very clean when the floor is being laid. Easier to do with tiles than it is with a huge sheet of vinyl.