Namaste,
thanks for the interesting OP.
What influence do you think the Christocentric religions like Nestorianism and Manicheanism had on the evolution of Mahayana Buddhism (specifically pure land, the messianic idea of the future Buddha Maitrya whose name is thought to be connected with Mithras/ Mihr , the Shamballah myth, and Tantric Buddhism)?
little to none. the principles of the Mahayana were established during the Buddhas initial Turning of the Wheel of Dharma. it is helpful to consider the idea that Hinyana, Mahayana and Vajrayana are merely useful mental coordinates to help a being navigate to the Other Shore.
i'm not a Pure Land adherent though Tariki, who frequenly visits this forum is and will be able to offer a more thorough insight into your question in it's specific context. i would offer by way of discourse that the idea of a Pure Land within the Buddhist context is far, far different than what is understood within the Christian paradigm as Heaven.
Centeral Asian peoples like the Sogdians, Turks, Uighers, Mongols , and Persians often practiced Manicheanism or Nestorian Christianity.
those peoples have practice a great many religious traditions which have arisen within their own socio-cultural paradigm or were imposed upon them by others so i'm unclear how this would have any sustained cultural impact especially given the diverse and geographically isolated societies of which those cultures are representative.
Central Asia is thought to have been the birth place of Mahayana by many scholars.
it is? which scholars think this? i would suggest scholars that hold such a view are astoundingly mistaken. one need but read the Suttas for themselves to find the teachings of the Yanas therein.
it is, however, often the case that the idea of categories, in this case groupings of the Buddhas teachings termed "hinyana, mahayana and vajrayana" represent some manner of difference in the teachings in the minds of a great many beings however this is not so.
one of the most overlooked aspects in the discussion regarding the Buddhadharma, in my experience, is the idea that what the Buddha taught isn't actually in the Suttas or Yanas, it's the Dharma and that Dharma is Suchness, Reality-as-it-is unadorned by mental obfuscation and delusion and this Dharma is something which is experienced outside of the texts.
Manicheans and Buddhists also practiced their faith in China for centuries. Eventually the Chinese Emperor issued an edict saying that only non-Chinese living in China could practice the faith because it was portraying itself as a school of Buddhism and confusing the simple.
Buddhists still do, insofar as the Chinese govt will permit them to. i'd be quite interested to read about the emperor that banned the Manichean practice.