Best idea I can recommend is to get yourself a mini manual, and have a look at the diagrams of how the front suspension goes together.
The only real differences between the wet and dry set up are:
1)Front: Top suspension arms have a different pivot point for the knuckle joint. They also need the shock mounting hole (where the bump stiop is fixed to on wet top arms) drilled out to suit the Shock mounting bolts. You can use the wet top arm with the dry set up with no worries, it is simply a firmer ride than if you used the dry top arm. Don't forget to refit the bump stop!!
2) Front: The wet subframe has mounting lugs in the top of the towers where the hydro bags 'lock' into. These can be modified so ther dry cones can sit in there. I think it would be better to use a dry subframe rather than modify a wet one, as once you mod it for dry there's no changing back. Since there are a lot more dry than wet subframes in the world, the logic is simple.
Call Ken Nelson at Mini Automotive, he might have a lead on some dry 'frames, and he is excellent for pretty much anything mini related.
3) Front, you will need to fit the shock mounts from a dry mini to the captive nuts up inside the guard. This means finding the holes first, cleaning out the crap, then using a tap 1/4" UNC I think, to clean up the thread. (This is also an excellent idea if keeping the wet as it will reduce the 'pitch' under acceleration. Fit rear comp bumpstops and that's even better)
4) Rear, subrames can be modded, as the front. Other changes will see you need to swap the trailing arms, for the dry ones with the shock absorber mount. You will need to change the rear brake line configuration to allow for the dry brake line style.
So, swapping the complete setup for another complete setup is definately the way to go IMHO.
On another note, you mentioned the coil over setup is what you will be going for. Isn't that a complete swap for dry anyway? So what you are thinking of doing is swapping from the wet setup to a coilover setup, not to a dry setup?
Good Luck
Matt