Had it been James the brother of John who wrote this epistle or had it been James of Alphaeus who wrote when James the brother of John was still alive (before execution by the second of the three Herods) then either of them would have had to make clear who they were.
That 'James' does not do so makes it clear that the epistle was written at a time when there was no ambiguity, hence it was written after the death of James the brother of John. And it was written by the surviving James - James of Alphaeus.
James the Lord's brother had some influence for a brief time, as Paul makes clear in the Galatian epistle, but with the Galatian controversy it became clear that the influence of James (brother of the Lord) was not reliable.
Thus no epistle of James the Lord's brother would have carried weight and would not have survived (as scripture) in the early church.
Thus the timing of the James' epistle is after the execution of John by Herod and in the period of the Diaspora (Acts 8) to which state the first epistle of Peter is also addressed.
My understanding is that Peter's First epistle 'unlocks' the Greek scriptures as he had the keys and it was his place, as the Chief Apostle, to write first. I would therefore say that I Peter is first, followed by Matthew and James and Jude (possibly in that order, possibly not quite).
The 'twelve tribes scattered abroad' are - at least - those scattered in the Diaspora but the very saying with which James opens his epistle (James 1:1) :
ταις δωδεκα φυλαις ταις εν τη διασπορα
the twelve tribes in the dispersion
Textus Receptus Interlinear
can be enlarged to cover any previous scattering and particularly the scattering of the ten tribes after the Assyrian onslaught, II Kings 17:6 :
In the ninth year of Hoshea the king of Assyria took Samaria, and carried Israel away into Assyria, [KJV 1769]
Thus I would see the epistle to be addressed to all living Jews of all twelve tribes, whatever their location and extant status. It is a very gracious inclusion, to my mind, and exclusive of none of the then living children of Israel whatever their own personal background history.
Both the content of the epistle (it's very similar expression to the prophets) and the timing of the epistle (after the execution of John but before the missionary expeditions of Paul) would indicate, to myself, that the church, as such, is not being referred to - it would be too soon.
This was definitely, to my mind, a transitional period and James addresses the situation as it was at the time, not as it would be, one or two decades later.