Just spotted the tread so will add my two-pennorth if I may...
Once clean, and it doesn't really matter too much IF you never use it again, what you use, I'd seal your wheels with a specialist wheel sealant, the Race Glaze Nano Wheel Sealant is well regarded. My £9.99 bottle is 3+ years old, half full and looks after 4 cars. Then wash with old mitt, brushes and water/shampoo if really dirty (drive car after to clear brakes, a few hundred yards with foot on brake will do) but given your circumstances, a selection of your mankiest microfibres and cheap quick detailer will dry clean them perfectly.
Sponge is big no no.
The idea of glaze using a filler-based product is excellent, any impact you can have by hand on swirls will be good but filler and 2 coats of good wax like the £68 RG55 mentioned earlier will have a magical effect. My well used (Hillclimb) 360 was on the FOC stand at the NEC 3 years back and wasn't machine polished, I used this same combo and no-one said anything.. Such a tub of wax will coat your car 30-35 times, lasts longer, shines better and fills better than cheaper stuff. It is worth the money IMHO.
Wash car from top first, as water running down will help soften dirt and mitt wont get caked from the start. Consider using old mitt for under sills. If using high quality wax (see above) then your shampoo suggestion will reduce the life of it and hasten the return of the dreaded swirls. You really must use something ph neutral to delay coating degradation.
A wash over and dry with soft drying towel (£6.99 - get 2) will do you on a regular basis, avoiding wheels. Wash MF cloths on their own without fabric softener which coats the fibres and wrecks them.
[posting now, will return when I can remember the rest of the thread...]
You'll need to re-wash the car after the claying process to remove the residue before you actually wax it.
If you want to use a wheel cleaner, then an iron-removing product like Kenotek Wheel Cleaner Ultra is great - you'll be able to see what you missed, and I can confirm from personal experience that even if it dries it washes off easily without staining alloys. But you'll need a fair amount of water to wash any wheel cleaner off, so I'd go with the sealant and QD cleaning route in your situation. If you can apply it when you pick up the car (if not at home yet) then that's the perfect time to do so. You'll not need to wet the wheels again.