Hi Alan,
I don't know if you've made a decision and commenced yet, or not? There seems to be a lot of confusing advice given (I know I'm confused by the replies).
Firstly, get all the vinyl tiles up.
Next, you need to asses how rigid your existing floor is and whether there is any deflection (bounce). If there is, then this needs to be dealt with using ply (18mm is recommended). When overboarding with ply, you should apply an acrylic primer (neat) to the underside and edges to seal it. Then stick the boards down with a flexible cement based adhesive (not no more nails / gripfill etc.!!), a 6mm trowel will be okay. Then screw the boards into the floorboards at 150 to 200mm centres. You will need to follow the adhesive maunfacturers instructions as to whether the face should be primed prior to tile fixing. Personally I use Ardex 7001 or Mapei Keraquick with latex when fixing to wood floors.
If there is virtually no deflection in the floor, you can use 6mm cement boards. Glue with a flexible cement based adhesive (6mm trowel) and then screw at 150 to 200 mm centres into the floorboards. You will then be able to fix the tiles using a single part flexible cement based adhesive.
I hope this makes sense.
Good luck.