Here's a wee piece from pushing my memory, since I'm not likely to feel like digging into theology this week. I've had enough of that recently, I reckon!
And besides, my memory of the Qur'an and the Hadith is not nearly as good as it should be, so it takes me a while to find what I'm looking for. Anyroad, it's just something for you to think about...
For every one of you [Jews, Christians, Muslims], We have appointed a path and a way. If G-d had willed, He would have made you but one community; but that [He has not donein order that] He may try you in what has come to you. So compete with one another in good works.
Surely they that believe, and those of Jewry, and the Christians, and those Sabeans, whoso believe in G-d and the Last Day, and work righteousness--their wage awaits them with their Lord, and no fear shall be on them, neither shall they sorrow.
The Qur'an is not clear on a theory of supersession and there have been arguments both ways in the legal traditions. The general consensus is that the Abrahamic faiths are to be left alone, with a variant of the old "Let G-d sort 'em out" mentality.
Other religions,
i.e. those not mentioned in the Qur'an as being of the Abrahamic lineage, are not so fortunate. It was to 'pagans' (non-Jews, non-Christians) that Islam was spread by the sword. Those of the other Abrahamic faiths in Muslim lands were tolerated and given religious freedom, so long as they paid their taxes and did not slander Islam.