Did the Emperor always have the correct legion in system when meeting a new Primarch?

Question

When reading of the reuniting of the Emperor with each of his Primarch sons it always seems that the Primarch linked up with his legion in a very short space of time.

For instance Angron seems to have joined the World Eaters very soon after being teleported away from death, Magnus instantly got on with trying to fix the Thousand Sons.

Given the size of the galaxy and the nature of the crusade I did think that maybe the Emperor kept all the legions without a primarch under his own command, but the Raven Guard fought largely under Horus before meeting Corax, hence the Earth based legionaries felt more loyalty to him then there primarch.

So are there any cases in the fluff of a Primarch taking a long time between being found and being united with his legion?

Answer 1

The Emperor was not even present at every discovery:

Like the other Primarchs, Alpharius and his twin brother Omegon were transported from the Emperor's gene-lab beneath the Himalazian (Himalayan) Mountains of Terra through the Warp by the Chaos Gods and placed on a far-away world in an attempt to prevent the coming of the Age of the Imperium and the expansion of the ordered Imperium of Man across the Milky Way Galaxy. Alpharius Omegon was eventually rediscovered by the Primarch Horus and sent to meet his father the Emperor on Terra, who placed him (and secretly, his brother) in command of the XXth Legion, the so-called Alpha Legion, during the Great Crusade of the late 30th Millennium.

From Fandom.Wiki

And with Corvus Corax, the Raven Guard was not present:

Whatever the truth, one thing is known to have passed between father and son that night. The Emperor would leave Corax to complete his mission and to defeat the tech guilds of Kiavahr. Only later would he be ready to assume command of his Legion.

Also from Fandom.Wiki

And Perturabo passed some time on earth before being given control of his legion:

After his rediscovery by the Imperium, the young Primarch was brought to Terra to learn from his father about his chosen destiny and to meet some of his brother Primarchs. During his studies, Perturabo learned of the ancient people known as the Firenzii. Captivated by the history of the past, Perturabo and his brother Magnus of the Thousand Sons Legion spent many months together in search of the buried secrets of Mankind's past glories that had been swept away in the chaos of the Age of Strife, known by Terrans as Old Night.

Once more, Fandom.Wiki