Russia indicated they would veto and both China and India were also opposed to it (though India doesn't really matter in terms of UNSC resolutions). Remember that Russia had been voting for earlier resolutions on the issue (1160 and 1199) and China had been abstaining to prior votes under the policy of nonintervention on internal matters.

It's important to note that, according to this translation of the Serbian Assembly, on March 23rd, the Serbian government had accepted the notion of Kosovo's autonomy under the condition of avoiding NATO intervention:

The National Assembly of the Republic of Serbia empowers the state delegation to sign a political agreement on self rule in Kosovo-Metohija on which agreement will be reached by representatives of all ethnic communities living in Kosovo-Metohija

(at the time, Albanians were still by far the dominant ethnic community).

I'd argue that NATO intervention was likely to be one of the key factors that stopped Russia from advocating for the agreement. With pressure from all other parties, it's hard to imagine a world where Serbia wouldn't be forced to come to a more agreeable compromise. The OSCE KVM prior to the deterioration of negotiations was doing it's job of preventing further escalation, after all, and it's important to note that only about a third of the 1.2 million Albanians displaced by the war had been displaced up to that point (largely due to work by the KVM) and that, as a direct consequence of NATO intervention, the forced displacement of Kosovar Albanians was accelerated significantly.