In the Lucifer FOX TV show, Lucifer's mother took the body of recently dead humans when she materialized on Earth. Whose physical body does Lucifer have while he is on Earth?
To the best of our knowledge, Lucifer's mother's requirement to take over someone else's body was unique to her. As far as we have seen, the bodies of the angels are their real physical form. A few things to note here:
- Lucifer has scars on his back where his wings were removed
- Amenadiel's wings are still attached and unfold from his shoulders
- The angels seem to recognize each other immediately (as when Uriel shows up), while Lucifer didn't think he would recognize his mother on sight.
All of that implies that angels bodies are their own, and thus they always look the same, while Lucifer's mom is incorporeal and needs to hijack a dead body to exist.
This makes sense when you consider the kind of entity that Lucifer/Amenadiel/Uriel/etc are, as opposed to God and Mom. The latter are cosmic entities -- the universe exists because they created it, so it seems self-evident that they have a form that exists "outside" of the world. The angels, on the other hand, were created as part of the world, so they would naturally have form like everything else in the world.
Updating the answer based on the latest season.
In season 4, it is mentioned that demons can only come to Earth by possessing a body. Similarly, we see Lucifer's mom taking over a body.
Essentially, all celestial creatures other than angels have no corporeal form. Therefore, if they wish to exist in our corporeal world, they need to take over a body.
The reason angels do have a corporeal form is because an angel's duty is almost solely defined by their Earthly interventions. Rather than try to hide from sight (Amenadiel could instead freeze time but the other angels have no such skill), they have a humanlike shape which allows them to roam the Earth freely.
On top of that, Lucifer is the one who outlawed demon possession, so it stands to reason that he wouldn't want to possess a body either.