Short answer
Not if the message of Islam reaches them and they reject it. The Qur'an promises reward to Jews and Christians who believed in God (﷾) and the Last Day and did good. Muslim scholars read this as praise for those who followed their own prophets before the Prophet Muhammad (ﷺ) came. After he came, following him became the way to God (﷾). The Qur'an says God (﷾) will not accept any religion but Islam from someone who seeks another. The Prophet (ﷺ) spoke of the Jew or Christian who hears of him and dies rejecting his message. He said that person will be among the people of the Fire. God (﷾) forgives any sin He wills except worshipping others beside Him. Believers who enter Hell for their sins will be taken out of it. And the Qur'an says God (﷾) punishes no one until a messenger has reached them. That judgement belongs to God (﷾) alone. Muslims declare no particular person's fate, and the Qur'an allows them to treat people of every faith with kindness and justice (60:8).
Full explanation
Before the Prophet (ﷺ) came
The Qur'an promises that the believers among the Jews, Christians and Sabians (a group the Qur'an names beside them) will have their reward. They must believe in God (﷾) and the Last Day (the Day of Judgement) and do good. No fear will be on them. Ibn Kathir, a scholar of the fourteenth century who explained the Qur'an, says this praises those who followed the messenger of their own time. A Jew who held to Moses until Jesus came was on the truth. So was a Christian who held to Jesus before the Prophet (ﷺ) was sent.
“Those who believe (in the Qur'an), and those who follow the Jewish (scriptures), and the Christians and the Sabians,- any who believe in Allah and the Last Day, and work righteousness, shall have their reward with their Lord; on them shall be no fear, nor shall they grieve.”
Explains that the verse praises those who followed the messenger of their own time: a Jew who held to the Torah and the way of Moses until Jesus came was a believer, as was a Christian who held to the Gospel and the way of Jesus; but once Muhammad (ﷺ) was sent, faith is accepted only from one who follows him. He cites Qur'an 3:85 for this rule.
After the Prophet (ﷺ) came
Once the last messenger (ﷺ) had come, following him became the way to God (﷾). The Qur'an says no other religion will be accepted from one who seeks it. A Jew or Christian may hear of the Prophet (ﷺ) and die rejecting his message. The Prophet (ﷺ) made it plain that this person is among the people of the Fire. The Qur'an says the same of the People of the Book (the Jews and Christians) who reject the truth once it has come to them. The door stays open. The Qur'an praises the People of the Book who believe in the revelation given to them and in the revelation given to the Prophet (ﷺ). It promises them their reward.
“If anyone desires a religion other than Islam (submission to Allah), never will it be accepted of him; and in the Hereafter He will be in the ranks of those who have lost (All spiritual good).”
Abu Huraira reports that the Prophet (ﷺ) swore that any Jew or Christian of his community who hears of him, then dies without believing in the message he was sent with, will be among the people of the Fire.
“Those who reject (Truth), among the People of the Book and among the Polytheists, will be in Hell-Fire, to dwell therein (for aye). They are the worst of creatures.”
“And there are, certainly, among the People of the Book, those who believe in Allah, in the revelation to you, and in the revelation to them, bowing in humility to Allah: They will not sell the Signs of Allah for a miserable gain! For them is a reward with their Lord, and Allah is swift in account.”
Christianity also names one way
Christians teach that salvation comes only through faith in Jesus Christ. They point to his words in the Gospel of John that no one comes to the Father but by him. They also point to the words of Peter, one of Jesus' disciples, that there is no other name by which people must be saved. Christian churches differ on those who never heard of Christ. So both faiths hold that there is one way to God (﷾). The question is which way is true.
“Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.”
“Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.”
The one sin God (﷾) does not forgive
The Qur'an names one sin that God (﷾) does not forgive: setting up partners with Him. Muslim scholars explain that this means one who dies on it without repenting. Every other sin He forgives to whom He wills. The Prophet (ﷺ) said that whoever dies holding that there is no god but Allah (﷾) will enter Paradise, even if he committed adultery and theft. Al-Bukhari explains this as one who repents at death or before it. The Qur'an reports Jesus himself telling the Children of Israel to worship Allah (﷾), his Lord and theirs. It reports him warning that whoever joins others with God (﷾) is barred from the Garden (Paradise). That is why belief about who God (﷾) is decides the matter.
“Allah forgiveth not that partners should be set up with Him; but He forgiveth anything else, to whom He pleaseth; to set up partners with Allah is to devise a sin Most heinous indeed.”
“Allah forgiveth not (The sin of) joining other gods with Him; but He forgiveth whom He pleaseth other sins than this: one who joins other gods with Allah, Hath strayed far, far away (from the right).”
Abu Dharr reports that the Prophet (ﷺ) said that anyone who says there is no god but Allah, and dies holding to it, will enter Paradise. Abu Dharr asked: even if he had committed adultery and theft? The Prophet said: even so, and repeated it each time Abu Dharr asked again. Al-Bukhari adds that this concerns the one who, at death or before it, repents and says there is no god but Allah: he is forgiven.
“They do blaspheme who say: "Allah is Christ the son of Mary." But said Christ: "O Children of Israel! worship Allah, my Lord and your Lord." Whoever joins other gods with Allah,- Allah will forbid him the garden, and the Fire will be his abode. There will for the wrong-doers be no one to help.”
“Those who reject Faith and do wrong,- Allah will not forgive them nor guide them to any way- Except the way of Hell, to dwell therein for ever. And this to Allah is easy.”
Mercy and justice
The Prophet (ﷺ) spoke of a person who bore witness that there is no god but Allah (﷾). He said that person will in the end be taken out of Hell. This is so even if he has only an atom's weight of good in his heart. The Qur'an also says God (﷾) punishes no one until He has sent a messenger to warn them. Muslim scholars hold that some people were never truly reached by the message. They are not judged like those who heard it and turned away. Muslims do not say that any particular person is in Paradise or in Hell. The only exceptions are those the Qur'an or the Prophet (ﷺ) has named. That judgement belongs to God (﷾) alone, who will weigh every atom of good and evil.
Anas reports that the Prophet (ﷺ) said: whoever says there is no god but Allah and has in his heart good the weight of a barley grain will be taken out of Hell; so too whoever has the weight of a wheat grain, and whoever has the weight of an atom.
“Who receiveth guidance, receiveth it for his own benefit: who goeth astray doth so to his own loss: No bearer of burdens can bear the burden of another: nor would We visit with Our Wrath until We had sent a messenger (to give warning).”
“Then shall anyone who has done an atom's weight of good, see it! And anyone who has done an atom's weight of evil, shall see it.”
Does 2:62 save everyone?
Qur'an 2:62 — Surat al-Baqara (The Cow), verse 62
“Those who believe (in the Qur'an), and those who follow the Jewish (scriptures), and the Christians and the Sabians,- any who believe in Allah and the Last Day, and work righteousness, shall have their reward with their Lord; on them shall be no fear, nor shall they grieve.”
Cited for
The Qur'an says Jews, Christians and Sabians who believe in God and do good have nothing to fear. So they are saved without following Muhammad (ﷺ).
The response
The verse itself sets conditions: belief in God (﷾) and the Last Day, and righteous deeds. Ibn Kathir explains that it praises those who followed the messenger of their own time, before the next one came. The Qur'an then says that no religion other than Islam will be accepted from one who seeks it. The Prophet (ﷺ) spoke of a Jew or Christian who hears of him and dies rejecting his message. He said that person will be among the people of the Fire.
Where it lands — 2:62 describes the faithful before the Prophet's (ﷺ) coming; after it, salvation means following him.
Perspectives
Mainstream Christian
Christians hold that salvation is by faith in Jesus Christ, who they say died for sinners and rose again. Churches differ on those who never heard of him.
John's Gospel has Jesus say that no one comes to the Father but by him. Christians read this as meaning that salvation is only through faith in him.
Rebuttal
Islam agrees that, in his own time, following Jesus was the way to God (﷾). Ibn Kathir explains that faith then meant following the messenger of one's own time. And the way Jesus taught was to worship Allah (﷾), his Lord and theirs (5:72).
In the Book of Acts, Peter, a disciple of Jesus, speaks to the council. He tells them there is salvation in no other name.
Rebuttal
Peter spoke before the Prophet (ﷺ) came. Then, faith meant following the messenger of your own time, as Ibn Kathir explains. Even then, Jesus called people to worship Allah (﷾), his Lord and theirs, not himself (5:72). Since the last messenger came, the Qur'an says no other religion will be accepted (3:85).
Jewish: the pious of the nations have a share
Judaism does not hold that only Jews are saved. Maimonides rules that a non-Jew who keeps the seven commandments given to Noah is one of the pious of the nations. He has a share in the World to Come.
A non-Jew who keeps the seven commandments given to Noah is one of the pious of the nations. He has a share in the World to Come.
Rebuttal
Maimonides adds a condition. The non-Jew must keep them because God (﷾) commanded them in the Torah through Moses. So Judaism too ties salvation to accepting the messenger God (﷾) sent. Islam says the same of the last messenger. Those who believed and did good under their own prophets have their reward. And God (﷾) punishes no one before a messenger has reached him.
The Islamic perspective
Salvation belongs to those who believe in the one God (﷾) and the Last Day, and do good, following the messenger God (﷾) sent to them. Since the Prophet Muhammad (ﷺ) came, the last messenger, that means following him.
The Qur'an promises reward, and freedom from fear, to Jews, Christians and Sabians. This is for those who believed in God (﷾) and the Last Day and did good.
The Qur'an says no religion other than Islam will be accepted from one who seeks it.
A Jew or Christian may hear of the Prophet (ﷺ) and die without believing in his message. The Prophet (ﷺ) said that person will be among the people of the Fire.
Some People of the Book believe in their own revelation and in the Qur'an. They have their reward with their Lord.
God (﷾) forgives every sin He wills except setting up partners with Him.
Believers who go into Hell for their sins will be taken out of it. This includes those with only an atom's weight of good.
God (﷾) punishes no one until a messenger has come to them.