There is an excellent book by R. G. Hoyland called "Seeing Islam As Others Saw It: A Survey And Evaluation Of Christian, Jewish And Zoroastrian Writings On Early Islam." that includes many primary sources from the first centuries of Islam with special emphasis on sources from Syriac, Armenia, Persian, Byzantine and Coptic sources on their contemporary reactions. It's been a long time since I've read it, but I can give a very general overview.

Late Antiquity at the time of Muhammad's rise and the expansion of Muhammad's empire was divided between two main "Superpowers" (using that not because it's a really accurate, but it will give a sense of how these two empires were viewed): the Byzantine Empire and the Persian Empire. Most of the other kingdoms in the area were under the sphere of influence of one of these powers or else desperately trying to stay neutral. The lands of the Fertile Crescent were divided and actual control, typically through vassal states, went back and forth between the Byzantines and the Persians. Muhammad's Mecca, however, was a backwater. While somewhat important regionally due to the trade routes, no major power was attempting to directly control this area though the Persians were fighting for control of the Yemen against the Kingdom of Aksum (based in modern Eritrea/Ethiopia) and some local dynasties.

What this means for Muhammad is that frankly nobody outside the immediate area around Mecca/Medina seems to have known about him or to have really cared. I'm about to leave for work in about 15 minutes so I don't have time to look up the sources and quotes, though I can try to find them later, but there very few sources that mention him directly in his lifetime and some of those that we do have might not be reliable. In general, once his empire started expanding in his later career, we start to have references to him in the chronicles of various monasteries (again, I don't have time right now to look up the exact ones, but I can try to get them later!). From memory, the main reactions were from the monks was that he was a Heresiarch, i.e. the leader of a heretical off-shoot of Christianity and this seems to have been quite wide-spread. The monks seem to have thought that he was a Christian who created his (in their view) false religion or else he was led astray by a heresiarch named Bahira (or Sergius in some sources), a corrupt monk who used the naive Muhammad to peddle his "revelation." However, this legend largely post-dates Muhammad and was used by later anti-Muslim Christian writers to discredit Muhammad. I'm not sure there's any evidence of it being contemporary.

Many Christian sources do talk about an expanding power in Arabia led by some sort of powerful leader but they are vague and contradictory on whether he and his movement are religious or just military and what exactly Muhammad's teaching are, although at least one Armenian source (again, I'll try to find this after work!) notes that Muhammad teaches that there is one god. Some Egyptian sources, who at the time were being oppressed by the Byzantine Emperor Heraclius, were more favorable to Muhammad. Heraclius was trying to force the non-conforming Coptic Christians of Egypt back into the Catholic fold (this is pre-schism) and the Arabs were greeted as liberators since they did not attempt (at the time) to convert the Christians, but left them in peace. However, the Muslim invasion of Egypt was in 639, 7 years after Muhammad's death. The Achtiname of Muhammad is a document that was supposedly given to the monks at Saint Catherine's Monastery in the Sinai in 626 by Muhammad offering freedom of worship to Christians. The "original" (if it is original) is in Instanbul.

Jews in the Holy Land were also more welcoming of the invading Muslims armies under Umar ibn al-Khattab in 634 (2 years after Muhammad's death). Umar signed treaties with the Christian and Jewish leaders granting them the right to worship freely. But again, this all post-dates Muhammad, though not by much. Jewish sources welcomed them as fellow monotheists opposed to the Christians and who, unlike the Christians, were willing to let the Jews worship freely. Keep in mind that both Christians and Jews had to pay the poll tax (the Jizya) that all non-Muslims had to pay as well as a land tax, but our modern notions of everybody equal before the law do not apply and the Jews were happy to be left in peace. There were, in fact, a number of Jews who followed Muhammad and who produced works in Arabic on the Old Testament Prophets collectively called the Isra'iliyyat (though not all writers of the Isra'iliyyat were former Jews).

This is a long way to say that we don't have a lot of information. Very few reliable contemporary sources directly mention Muhammad. Most of what we have post-dates the later expansion of the Arab Kingdom into the Fertile Crescent, the Levant, and Egypt after Muhammad's death in 632 AD and there are no surviving non-Muslim sources from Mecca and Medina. Christians outside of Arabia tended to view him as a military leader making raids (as Arabs had from time immemorial) on trade caravans and non-conforming Christians at least accepted his followers rule as an improvement over the oppressive Byzantines.

A separate case would be the Christians of Ethiopia with whom the early Muslim Community sought refuge. We only have Muslim sources on this, but it is clear that many Muslims fled the persecution of their early days in Mecca by fleeing to what is now Ethiopia. According to legend, the Negus (King) listened to a delegation of Muslims and a delegation of pagan Meccans denouncing the Muslims and recognized the Muslims as fellow monotheists and granted them safe haven. Again, we only have Muslims sources on this. In Arabia there were many Christians and Jews. Some Jews survived in Yemen until the modern era, but all the Christian Arabs eventually converted.

I can add more later, but sadly I have to go to work!

Edit: a few words. I'm at work now. I'll try to add more details and quotes later.