In Luke 5 we have the conversion of Levi. It appears that he became an apostle almost straight away. Is that what is happening here?
Yes. From that point on Levi became a disciple of Jesus.
Levi left everything and followed Jesus. The Greek for left everything, katalipon panta, means abandoned all, forsook or deserted. The word for followed is in the imperfect tense, not a one-off action but a continuing state.
He didn't knock off for the day, he quit his job. He didn't sneak off early, he abandoned his post. That evening there was a party at his home to which Jesus was, of course, invited. Many other tax collectors were present. It was in part a Leaving Do in which many of his former colleagues came to say goodbye. It was also a chance for them to meet Jesus, and to understand for themselves what had prompted his resignation, and the new life he had chosen.
It was not until some time later, perhaps a few weeks, that Jesus chose from amongst his many disciples Twelve in particular. So whether or not Levi became one of the Twelve it did not happen immediately. Long tradition says that Levi and Matthew were one and the same person. St Augustine of Hippo in his Harmony of the Gospels Book 2 Chapter 26 thought so. In Matthew, Mark and Luke the story of the calling of a tax collector immediately follows the healing of the paralytic, and all say the calling of the tax collector was followed by a meal at his home, which led to identical complaints from the pharisees. It is widely conjectured that this was one and the same incident.
Luke and Mark refer to the tax collector as Levi, and include the name Matthew amongst the Twelve. If he is one and the same they may be referring to "Levi" as the name he used then, rather as we might refer to an incident involving the Duchess of Cambridge before her marriage as happening to a Miss Middleton. They may also have wished to avoid embarrassing Matthew by mentioning his former profession. Matthew, in his own gospel, refers to the tax collector as Matthew, modestly having no wish to hide his shady past. Similarly, whereas Luke refers to a great banquet, Matthew plays down the occasion as a dinner.
Following the Resurrection Matthew became one of the Apostles, as did all the Twelve disciples apart from Judas. The difference between a disciple and an apostle is that a disciple learns and an apostle teaches. During Christ's earthly ministry they were disciples, afterwards apostles.
So yes, Levi became a disciple immediately. He probably, though some reasonably doubt it, became one of the Twelve Disciples when they were called, a few weeks later. If so, after the Resurrection he became an apostle.
All three Gospel accounts of Levi's conversion are consistent: they do not mention anything intervening between Jesus' telling him to follow and his rising and following Him:
Matthew 9:9 (RSV)
As Jesus passed on from there, he saw a man called Matthew sitting at the tax office; and he said to him, “Follow me.” And he rose and followed him.
Mark 2:13–14 (RSV)
He went out again beside the sea; and all the crowd gathered about him, and he taught them. And as he passed on, he saw Levi the son of Alphaeus sitting at the tax office, and he said to him, “Follow me.” And he rose and followed him.
Luke 5:27–28 (RSV)
After this he went out, and saw a tax collector, named Levi, sitting at the tax office; and he said to him, “Follow me.” And he left everything, and rose and followed him.
@davidlol has explained why Matthew is called "Levi" in Mark's and Luke's account, but "Matthew" in his own.
Levi never becomes a disciple or apostle, unless we assume that he is also called Matthew, but Luke's Gospel never makes this assumption.
The passage in Luke 5:27 is known to have been sourced from Mark, which mentions both Levi and Matthew separately, referring to Levi, son of Alphaeus, as a tax collector whom Jesus called to follow him (Mark 2:14). As with Luke, Mark never again refers to Levi, who is not mentioned in the full list of the twelve disciples (verses 3:14-19), where Mark introduces other disciples including Matthew, Thaddaeus, and James, son of Alphaeus. In this list, we do have a son of Alphaeus, although this disciple's name is James, but we do not have a tax collector (publican). In this gospel, as in Luke's Gospel, there is no good reason to see Levi and Matthew as being the same person.
Levi arose and followed Jesus in Mark 2:14 and Luke 5:28, but is never again mentioned in either gospel, leaving open the possibility that he changed his mind. Disciples are not meant to change their minds when called by Jesus, but it is only Matthew that recognises this issue and creates a solution. Its anonymous author resolves Levi's unexplained absence among the disciples simply by not mentioning Levi at all, and by having Matthew as the disciple who was a tax collector, in the parallel account at Matthew 9:9, then including him as "Matthew the publican" in the list of disciples at Matthew 10:3. Almost two thousand years of tradition have followed Matthew in holding that Levi and Matthew must be the same person. But this happens only in Matthew's Gospel, not here in Luke.