You basically use the regular altimeter for altitude during such approaches. It's accurate enough to get to the decision height of a few hundred feet AGL after which you have to fly the rest of the approach manually anyway. That is true with CAT I ILS as well.

A CAT III ILS approach requires very specialized equipment and training because you basically fly it all the way down to the runway with no requirement for visibility until moments before touchdown.

However, GPS is a lot better when it is not being obstructed by buildings around you. Planes have the benefit of unobstructed paths to GPS satellites and would get a better quality and accuracy then you might get with a phone in a city. Also, using good antennas helps. So, the altitude accuracy would be fine mostly.