Annam will only return when his sons restore the glory of Ostoria.
According to Giantcraft p.44, Annam left Toril thousands of years ago and promised his wife Olthea that he would never return until his sons restored the glory of Ostoria. He has so far kept this promise and never returned. The 5e Monster Manual p.153 relates that Annam forsook the world and the giants when Ostoria fell, and disowned them until they returned the world to the rule of the giants.
According to Giantcraft p.10, Ostoria will return after a great war among the humanoids:
Olthea knew immediately that the son she carried in her womb was sired by Annam; she initially refused to bear the child. Eventually, she and Annam ended their struggle in a stalemate: In return for the All-Father’s promise to turn his back on Toril until the day the child called his name, Olthea promised not to expel her unborn child prematurely and allow the ogres to feast on its flesh and blood. Although the compromise would delay the child’s birth for a very long time, this delay became part of Olthea’s plan. …
Her strategy for rebuilding Ostoria looked forward to an era in which these new inhabitants [the humanoid races] would war amongst themselves, depleting their strength. Only then would the giants’ king emerge to rebuild Ostoria and restore the glory of the colossal kingdom.
This son was Hartkiller, who was slain in the War of the Hart.
Giantcraft p. 12 discusses the events leading to the future restoration of Ostoria:
Just after the War of the Hart ended, all the giants in the vicinity of Hartsvale were visited by a mysterious being of purple mist. The presence told each of the giants that by refusing to accept Hartkiller as their king, they rejected the will of Annam. The new kingdom of humans would serve as their punishment and a constant reminder of their insolence. Eventually, a young king born of Hartsvale’s royal line would claim Hartkiller’s legacy and re-establish Ostoria, paving the way for Annam’s return.
The mysterious giant figure appears on an annual basis, and there is speculation that it is in fact Annam in disguise.
Giantcraft p.47 says that when Annam returns, his son Stronmaus will relinquish the the rulership of all giants to him.
According to Powers and Pantheons p.62, Annam’s return would have significant ecological consequences, melting the icy northern region:
Ulutiu has not manifested an avatar in the Realms in almost 4,000 years and has no intention of doing so in the forseeable future. However, if the eternal rest of his final avatar entombed at the bottom of the Great Glacier was disturbed, if someone were to discover a method by which Olthea could be resurrected, or if Annam was ever to manifest an avatar in the Realms, Ulutiu would promptly manifest an avatar once again. Such an appearance would probably presage the dissolution of the Great Glacier, although that change would take centuries to occur.
Volo’s Guide to Monsters p.31-32 says that Annam’s return will be heralded by the birth of the greatest giant yet, as told in a system portents called the Ordning which ranks giants’ social order. Annam’s return will upend the social order of the giants, and bring about a new age. A great war may break out between those who benefit under the new order, and those who oppose it.
In the Tyranny of Dragons storyline, Annam has dissolved the Ordning to rouse the giants from their complacency, bringing about an era of conflict among giantkind as they fight to establish status. This is described in the adventure module Storm King’s Thunder.