Given what and where they were from it almost certainly means donated or sold to the museum by the archeologists who excavated them in the first place.
"From 1877 to 1882, while undertaking four expeditions on behalf of the British Museum, Rassam made some important discoveries. Numerous finds of significance were transported to the museum, thanks to an agreement made with the Ottoman Sultan by Rassam's old colleague Austen Henry Layard, now Ambassador at Constantinople, allowing Rassam to return and continue their earlier excavations and to 'pack and dispatch to England any antiquities [he] found ... provided, however, there were no duplicates.' A representative of the Sultan was instructed to be present at the dig to examine the objects as they were uncovered."
Yes but can everyone involved - both the original archeologists who excavated them and the people who bought them - claim ancestry from whatever ancient society made them, or at least the oldest traceable group to have resided on that land? AIUI that is how "stolen" tends to be construed for archaeological things these days.
That idea makes sense in an American context where there is a clear divide between indigenous and more recent groups. The same is true of some other places like Australia.
In Eurasia and Africa this does not work. "The oldest traceable group" may be many invasions, conquests and mass migrations after the things were made. The current residents or earliest traceable group may well have enslaved, oppressed, or killed the group who made the things. In many, many cases they would have erased the culture that made them.
In the case of these tablets there is no one who can meaningfully claim to be an Akkadian.
Given what and where they were from it almost certainly means donated or sold to the museum by the archeologists who excavated them in the first place.