It is my understanding that the reason God needed to deliver a teaching to Mohammed in the first place is because the Jews and Christians irredeemably strayed from the teachings of Moses and Jesus.
Does Islam have an opinion on approximately when
to the extent that another prophet was needed?
Yes. This is one of the things that is thoroughly explained in Quran, in many verses.
To make it short, both absolutely went wrong when they denied the prophet Muhammad as a prophet, since both Jews and Christians teachings has told that there was to be another prophet coming. This has already indicated that their teaching was not yet complete until the final prophet has come. So, in other words, the prophet Muhammad was sent as a prophecy not because the Jews and Christians went wrong, but because the teaching was not yet complete.
If they hadn't denied the prophet Muhammad when he came, certainly they would be forgiven of previous wrongdoings. Two notable examples of each wrongdoings and hence gives a vague start of when they turned wrong:
Reference:
If you read Quran, it is mentioned that Jews go wrong when they disbelieved in Jesus Christ (pbuh) and they still think that Jesus didn’t came to them and they think that Jesus will come one day, and as for Christians, they disbelieved when they committed shirk. And when Jews heard that prophet Muhammad will come, they tried to harm him when he was young according to Quran.
And that is how People of Book go wrong.
If you see something wrong, you have the right to edit my answer.
That assumption is wrong since 'irredeemably strayed' is assuming, they are all not believers. And Quran says that even Christians and Jews will go to paradise. 2:62
Indeed, those who believed and those who were Jews or Christians or
Sabeans - those who believed in Allah and the Last Day and did
righteousness - will have their reward with their Lord, and no
fear will there be concerning them, nor will they grieve.
Please note that Allah did discriminate no group when He gave humanity the Quran. Logically speaking everyone can commit sins of another 'ethnic group', specially when muslims became christians or jews and so on.
Islam can mean Peace accquired by Submiting yourself to Allah, in that context, Quran is the ultimate vehicle for achieving that peace. It is the truth for the belivers and not an opinion.
Note: I assume here that muslims are the most perfect form of believer (muslim can mean a person who submits to Allah). Many people of other ethnic groups can be muslims (and they don't know).
But they have attributed to Allah partners - the jinn, while
He has created them - and have fabricated for Him sons and
daughters. Exalted is He and high above what they describe
The Jews say, "Ezra is the son of Allah "; and the Christians
say, "The Messiah is the son of Allah ." That is their
statement from their mouths; they imitate the saying of those
who disbelieved [before them]. May Allah destroy them;
how are they deluded?
You will surely find the most intense of the people in animosity
toward the believers [to be] the Jews and those who associate others
with Allah ; and you will find the nearest of them in affection to
the believers those who say, "We are Christians." That is because
among them are priests and monks and because they are not arrogant.
Do you covet, that they would believe for you while a
party of them used to hear the words of Allah and then distort the Torah after
they had understood it while they were knowing?
And when a messenger from Allah came to them confirming that which was with them,
a party of those who had been given the Scripture threw the Scripture of Allah
behind their backs as if they did not know [what it contained].
Indeed, those who disbelieve in Allah and His messengers and wish to
discriminate between Allah and His messengers and say, "We believe in
some and disbelieve in others," and wish to adopt a way in between -
Those are the disbelievers, truly. And We have prepared for the disbelievers
a humiliating punishment.
(Targets Children of Israel)
Cursed were those who disbelieved among the Children of Israel by the
tongue of David and of Jesus, the son of Mary. That was because they
disobeyed and [habitually] transgressed.
You see many of them becoming allies of those who disbelieved.
How wretched is that which they have put forth for themselves
in that Allah has become angry with them, and in the punishment
they will abide eternally.
Note: Muslims have this reminder. (5:51)
Indeed, We sent down the Torah, in which was guidance and light. The
prophets who submitted [to Allah ] judged by it for the Jews, as did
the rabbis and scholars by that with which they were entrusted of the
Scripture of Allah, and they were witnesses thereto. So do not fear
the people but fear Me, and do not exchange My verses for a small price.
And whoever does not judge by what Allah has revealed - then it is
those who are the disbelievers.
