This is quite difficult to answer, as outside of the Qur'an and Muslim traditions from the late eighth century we have nothing describing what exactly happened in Mecca in the early seventh century. There is no reason however to ignore all later traditions, since they obviously must have preserved some aspects of what actually happened, and the Qur'an itself can tell us a lot about the world of Muhammad. It is nonetheless worth emphasising that we can't quantify the speed of conversion or the motives behind it, certainly not to the level of detail you might have expected. We can make educated suggestions, but nothing can be known for certain about this enigmatic but fascinating period of history.

Following the traditional narrative, Muhammad's revelations quickly attracted interest from his friends and family, with the earliest converts coming from all sorts of backgrounds, including slaves and wealthy merchants. However, it is also true that the list of the earliest converts preserved in sources such as Ibn Ishaq reflect the political realities of the eighth century, since whether someone was the descendants of the earliest converts or not became quite important in the later period. But whilst the order of conversion might be debatable, I think it is quite clear that many Meccans were convinced by Muhammad's teachings, though we always have to be aware at every juncture that we shouldn't take later sources at face-value. The Qur'an also suggests that the message was not well-received by some and that this led to the Hijra, or Emigration, to Medina. Muhammad's teachings probably ruffled some feathers in Mecca, as his message can be read as a warning to his community about the need to purify themselves (AKA to stop sinning against God and to abandon polytheism, though monotheism of various stripes must have been present in Mecca or at least known of there), so we can see Muhammad as some sort of an apocalyptic prophet (which fits very well into the milieu of the time) and thus an undesirable upstart for anyone with a vested interest in the status quo.

Muhammad was (traditionally) said to have been invited to Medina to mediate between different interests in the town, which suggests that Muhammad was reasonably successful at disseminating his message beyond Mecca already. As Muhammad was no doubt a charismatic and able man, I see no reason why we should discount this. It is however worth noting that Muhammad was not just a prophet, but a shrewd political leader as well, so we shouldn't discount the political aspects to the growth of the early Muslim community. Politics and religion cannot be separated and surely there is no doubt that Islam gained its adherents both through genuine faith but also because of the very real political/social/cultural need the new movement fulfilled. Medina is a case in point, since it is alleged that its warring clans needed an outsider to arbitrate its internal disputes, whilst the inclusion of the Jews in the Constitution of Medina suggests that Muhammad's message had some appeal to Jews as well. If we follow Fred Donner's recent argument, the early community of Believers was not just limited to Muslims, but one fully inclusive of Jews and Christians too, with their mutual goal being the end of polytheism. Even if we don't follow Donner, the early Muslim community clearly did tolerate the various Peoples of the Book and in doing so appealed to the religiously diverse inhabitants of Arabia, since Christianity was well-entrenched in Northern Arabia, and Judaism and Christianity in the south; there were even smatterings of Zoroastrians who lived on the eastern and southern coasts of Arabia.

Soon after his arrival in Medina Muhammad began a campaign against the Meccans and their allies, which after many twists and turns eventually led to his re-entry into Mecca from a position of strength (or of compromise, it depends on how you read the evidence). In the process Muhammad seemingly gained many allies and/or submission from tribes across Arabia, further increasing the reach of his teachings. It is debatable whether this led to widespread conversion or whether they were genuine if conversions happened though, since many people seemingly apostatised against Islam after Muhammad's death. There are also hints that Muhammad sent a force into Palestine as part of his campaign to secure the Holy Land (an Abrahamic apocalyptic movement of course had every reason to secure their rightful inheritance before the end times), which can be a signal to the sheer power of Muhammad's movement in the 630s. This however may be a garbled account of later Arab attacks misattributed to Muhammad, and more generally speaking Arab raids into Rome/Persia had been escalating since the 600s, so this is rather debatable.

Even ignoring this however I think it is pretty clear that Muhammad was a man of many talents, who secured the allegiances of many Arabs through both faith and politics, two intertwined concepts that allowed him to secure the nominal allegiances of people across Arabia as well as the devout belief of many, especially amongst those who had suffered through the darkest hour of the early community alongside the Prophet. If you are interested in what happened afterwards, you might like to read my comments on why the Arab Conquests succeeded and early Christian reactions to Islam; within decades, the confederation forged by Muhammad would destroy one world empire and shatter another, transforming utterly the world of late antiquity, a telling indication of Muhammad's skill in binding his followers together and in building a new polity out of a divided Arabia.

Further reading:

New Cambridge History of Islam, Volume I (2010) - an excellent summary of what modern scholars think about Muhammad and early Islam; the study of this period has transformed utterly from the 60s/70s, so it really is necessary to start with the most recent books possible.

F Donner, Muhammad and the Believers (2010) - a book written for the general public and so very readable, but presents a few interesting ideas. Read this review by Patricia Crone as well to balance out what you read in Donner's book.

H Kennedy, The Prophet and the Age of the Caliphates (2004 edition) - the standard book on all of this, not too critical of the sources but should give you all the basics.

B Sadeghi, 'The Codex of a Companion of the Prophet and the Qurān of the Prophet', Arabica, Volume 57/4 (2010) - a very technical article, but I think it establishes quite convincingly that the Qur'an, or at least a version of the Qur'an, can be dated back to Muhammad's lifetime.