No, James makes clear the "lost" tribes are were not destroyed by the Assyrians and had made it to wherever he was teaching (around Greece or Rome probably).
Jam 1:1 James, a servant of God and of the Lord Jesus Christ, to the twelve tribes which are scattered abroad, greeting.
The Assyrians took the "lost" tribes and relocated them East, while bringing in peoples from the East to occupy the land that was once the home of the "lost" tribes. In much the same way the Scots made excellent warriors for the British Empire, the "lost" tribes were excellent warriors for the Assyrians and they used them as a buffer on the border with Media-Persia, thereby gaining their freedom from captivity in a hurry. The Assyrians called the tribes the Bit-Sakae (house of Isaac) and Bit-Kumri (house of Omri, or Homri, with a hard "H"), and once they started moving west they were known as the Scythians and Cimmerians, and later mainly by the name Saxons. Sons of Issac = Bit Sakae = Scythians = Saxons. Genesis confirms that the descendents of Isaac would be called by the name of Isaac:
Gen 21:12 And God said unto Abraham, Let it not be grievous in thy sight because of the lad, and because of thy bondwoman; in all that Sarah hath said unto thee, hearken unto her voice; for in Isaac shall thy seed be called.
They then migrated north and then west (being close-by to be taught by James in the first century AD) and then continuing West and North into Northwestern Europe and the UK countries and America. It's no accident that Americans and British have names from the bible (Michael, James, John, Gabriel, Steven, etc., etc., etc.), we are descended from those that had the names in the first place.