If there were a natural progression from Christianity to Krishna Consciousness, then the Pope would be a Krishna bhakta or the evangelicals of the Bible belt would be filling ISKCON temples in America. In the 1960s and 1970s, young men and women who were dissatisfied with their nation, church and family were attracted to Hindu and Buddhist masters. They did not want to be atheists or materialists, but found the doctrine of our earthly suffering due to Adam eating an apple (Original Sin), bloody human sacrifice of Christ to eliminate the sin of eating an apple (Atonement) and eternal banishment to Hell for the people who did not believe the childish doctrine (Eternal Damnation) an insult to their reason. So, they joined many Oriental masters. Of course, they did not want Jesus to be insulted because he was a gentleman compared to his times. The Oriental masters also recognised that case and praised Jesus, who had taught to turn the other cheek or only the sinless should throw the first stone. He seemed a Buddha-like character, challenging the organised dominance of the priests and animal sacrifices in the Temple. Then, he fearlessly embraced his death and asked God to forgive those responsible. So, he looked like a great man. There was nothing wrong in praising him and the masters accommodated him.
However, in the contemporary Occidental societies, it has become fashionable to be atheist, materialist or quasi-atheist-materialist (New Ager), so fewer young men and women are turning to the East. The people attending ISKCON temples are mostly Hindus who have attraction for Krishna, not Jesus. Krishna Consciousness being a strictly traditional religion is unable to acquire the customers as much as the easy-going, hodgepodge New Agers; therefore, some devotees have inaugurated the Krishna West to get customers who are Hinduphobic, but want to enjoy some aspects of Hindu culture and philosophy. This process is called digestion. Just take yoga, vegetarianism and reincarnation and abandon strict sadhana and sense-control, especially that of visiting backward, dirty and criminal infested Vrindavan and Mayapur. Thus, while the earlier generations of Occidental devotees admired India, served in India, learnt from India and supported the regeneration of Krishna Consciousness in India, the Krishna West wants to capture a different consumer base. They want all the cool stuff that sells including Jesus Christ, meek as a lamb, but flush down the drain all the 'Hindu' stuff. This process is called dilution.
All of us not born in Vaishnava families have their own family gods and goddesses. We respect them, but finally have to discard them to take the exclusive shelter of Krishna, ekam sharanam. So should be the case with Christians.
If you are following the trend, what is going on in the West is digestion and dilution, whether it is Krishna West or Hinduism. Why was Oxford Centre for Hindu Studies established? Who are the leaders of Hindus in the UK? Who represents Hinduism in inter-religious dialogues? What Krishna West has done is to identify certain elements of Krishna Consciousness as Hindu and made a case for abandoning them. So, are we Hindu or not? For donations and recognition, we are Hindus; but so far as socio-cultural issues are concerned, we are not Hindus. This is a serious identity crisis.
A positive dialogue has to include genuine criticism, whether of Christianity or Hinduism. If KC is not Hinduism, it is not Christianity either. Indeed, Hinduism is just a generic term, it does not represent any particular faith, although many scholars are attempting to formulate a coherent superstructure for Hindusim (for instance, Rajiv Malhotra in Indra's Net: Defending Hinduism's Philosophical Unity http://indrasnetbook.com/). On the other hand, Christianity already has an agreed doctrine, i.e., the Nicene Creed:
We believe in one God,
the Father, the Almighty,
maker of heaven and earth,
of all that is, seen and unseen.
We believe in one Lord, Jesus Christ,
the only Son of God,
eternally begotten of the Father,
God from God, Light from Light,
true God from true God,
begotten, not made,
of one Being with the Father.
Through him all things were made.
For us and for our salvation
he came down from heaven:
by the power of the Holy Spirit
he became incarnate from the Virgin Mary,
and was made man.
For our sake he was crucified under Pontius Pilate;
he suffered death and was buried.
On the third day he rose again
in accordance with the Scriptures;
he ascended into heaven
and is seated at the right hand of the Father.
He will come again in glory to judge the living and the dead,
and his kingdom will have no end.
We believe in the Holy Spirit, the Lord, the giver of life,
who proceeds from the Father and the Son.
With the Father and the Son he is worshiped and glorified.
He has spoken through the Prophets.
We believe in one holy catholic and apostolic Church.
We acknowledge one baptism for the forgiveness of sins.
We look for the resurrection of the dead,
and the life of the world to come. Amen.
One who does not believe in the Nicene Creed is not a Christian. The interesting case may be someone who believes in both the Nicene Creed and Krishna Consciousness. I wonder if Shrila Prabhupada intended such syncretism. In that case, we also have to accommodate Muslims for whom Nicene Creed is haram. Muslims adhere to Kalima:
There is no god but Allah, [and] Muhammad is the messenger of Allah.
I bear witness that (there is) no god except Allah; One is He, no partner hath He, and I bear witness that Muhammad is His Servant and Messenger.
Exalted is Allah, and praise be to Allah, and there is no deity except Allah, and Allah is the Greatest. And there is no might nor power except in Allah, the Most High, the Most Great.
(There is) no god except Allah - One is He, no partners hath He. His is the Dominion, and His is the Praise. He gives life and causes death, and He is Living, who will not die, never. He of Majesty and Munificence. Within His Hand is (all) good. And He is, upon everything, Able (to exert His Will).
I seek forgiveness from Allah, my Lord, from every sin I committed knowingly or unknowingly, secretly or openly, and I turn towards Him from the sin that I know and from the sin that I do not know. Certainly You, You (are) the knower of the hidden things and the Concealer (of) the mistakes and the Forgiver (of) the sins. And (there is) no power and no strength except from Allah, the Most High, the Most Great.
O Allah! I seek protection in You from that I should not join any partner with You and I have knowledge of it. I seek Your forgiveness from that which I do not know. I repent from it (ignorance) and I reject disbelief and joining partners with You and of falsehood and slandering and innovation in religion and tell-tales and evil deeds and the blame and the disobedience, all of them. I submit to Your will and I believe and I declare: There is none worthy of worship except Allah and Muhammad is His Messenger.
There have been Muslim devotees of Krishna also. Damodar Nityananda Prabhu is coming out with a book on the subject: https://www.facebook.com/photo.php?fbid=10101829384231146&set=a.10101829384315976.1073741828.1531324&type=1&theater
Moreover, both Nicene Creed and Kalima would be unacceptable to the Hindus. Therefore, we need not create absurd categories like Christian Vaishnava or Muslim Vaishnava and provide authentication certificates to the traditions of Abraham, which already dominate the civilisational space of the world.