No Old Testament, or only limited bits, means that Christianity and Judaism are very separate from the get-go. This may affect the relationships between Islam and Christianity, as Islam "accepts" much of the narrative of the Old Testament albeit with alterations and omissions. Rather than seeing Christians as people of the book, they may be treated the same as those who are not of the Abrahamic tradition. This would have major Mothra size butterflies. Excluding, rather than superseding/modifying the Old Testament, will make conversion of Jews to the new faith much more problematic and potentially relationships between Christians and Jews even worse than OTL.