Hoffnung:
Note that God is going to "gather all things together again." (NWT) "Gather ... together again" is the NWT translation of anakephalaioo. The ana prefix means "again," and the rest means "to sum up, recapitulate. To sum up, gather together again in one. In the mid[dle] voice (as in Eph 1:10), to reunite under one head, to bring all things into one in Christ" (AMG Greek-English Dictionary, p. 152)
It is also used in the passive voice in Rom 13:9, where the NWT translates it "summed up."
Most commentators take "the things in the heavens ... earth" as a reference to all creation which has been 'disturbed/left in chaos' with the outbreak of sin.
The Society says "the things in the heavens" refer to the 144,000 and the "things in the earth" to earth-bound believers starting with the great crowd.
This raises several questions:
Why would Paul refer to Christians on earth (in his time) as the things "in the heavens"? None of the 144,000 were in heaven at that time.
Why would he refer to them in the neuter (the things"), when they were all people?
Why would he refer to the 144,000 as being "gathered ... together again"? The idea of a 144,000 only became a necessity after sin broke out. There was no 144,000 dispersed by sin that needed to be "gathered ... together again."
Why would the context (Eph 1:11 - 14) include believers as an additional thought ("in union with whom we were also assigned" [vs. 11] and "but you also" in vs. 13) if they were already referred to as "the things in the heavens."?
Like you said, there is no expression of "stages," although one might argue that God would accomplish this in an orderly fashion.
This idea of unifying the cosmos under Christ is also alluded to at 1 Cor 15:28 (where all things are subjected to the Son, and the Son to God), Phil 2:10 (where every creature - heavenly, earthly and subterranean - pays homage to Jesus for God's glory), and especially Col 1:20 (where God reconciles all things, whether in heaven or on earth, to himself through Christ). BECGNT, Ephesians, p. 67
Romans 8:18 - 25 also expresses a similar thought with "the things in the heavens ... earth" being replaced with "the creation."
(In Col 1:21-23, after mentioning "the things in the heavens ... earth" in verse 20 there is a similar addtional ("indeed" [NWT] which is kai in Greek, normally translated "and") idea expressed about believers. They believed the good news which was preached "under heaven." So it would be odd to refer to them as "the things in the heavens" and in the next clause as hearing the preaching "under heaven.")
Take care