I see it as a preposition. Here’s why:
Subordinating conjunctions function as markers of subordination, whereas preps (inc "after") function as heads of the constituents they introduce.
Unlike subordinating conjunctions, preps have independent meaning ('an evident semantic content'). In the case of "after", it has a temporal meaning.
With items like "after" they uncontroversially occur as preps when they have an NP as complement, and there's no basis for assigning them to different categories according as they take an NP or a clause - or no complement at all.
Trad grammar has:
after the meeting: preposition + noun
after we arrived: subordinating conjunction + sub clause
I didn’t seen her after: adverb, no complement
This is just a matter of varying complementation, which is commonplace.
I know her father: verb + NP
I know that he's ill: verb + sub clause
I know: verb without complement
a belief in God: noun + PP
the belief that God exists: noun + clause
her beliefs: noun without complement
Moreover, in all three constructions, "after" takes the same modifiers, e.g. a short while. We need therefore to distinguish "after" from the subordinators "that/whether", and then once we've done that "after" clearly belongs with all the other preps. For the record, Jespersen argued for treating "after" the same in all three constructions nearly a hundred years ago.