I'm not so sure those were ever polished? Weren't they more of satin finish when new?
I'd start with acid and a scotchbrite pad like guys said. Things I have used to clean aluminum:
You can use aluminum cleaner that the auto paint store sells.
Or you could use "coil cleaner" that HVAC guys use to clean coils.
Or I have some alkaline degreaser that will etch dirt out of aluminum just like the above products.
Or probably easiest, mag cleaner that etches like RR said. I don't think you want to polish?
If they were polished originally, then I would use a wool or cotton buff on them after the acid. A foam ball is very weak compared to a cotton buff.
I second the Autosol, I have done shitloads of polishing in the past and from all the polishes I tried, Autosol is the best, imo.
I could completely polish that simple wheel in less than 10 minutes per wheel if it was originally polished with my polishing tools.. Up to 2 hours per wheel would be crazy in my mind, sorry.
The red stuff in the spray bottle is the alkaline degreaser that will lift dirt out of aluminum and etch it. It's made by Johnson Wax and called Job Shop 600. Sold to strip waxed floors and is a awesome degreaser.
Use the red stuff to get brake dust off of aluminum wheels. It'll etch the polished, got to keep it away from polished aluminum, but boy does it clean. If it get's on the polished, I can buff it it out and make it look new again.
Wheel on the left is the inside wheel and not polished, has a satin finish that I have restored many times since 2002 with just the degreaser and a pressure washer.
I polish tanker trucks at work sometimes and buy it by the case.