It is extremely tricky, because it could be argued that Christianity is essentially the form Judaism took when it had to spread the Gentiles.
However, here is my (pretty unplausible) attempt.
The Khazar conversion to Judaism is supported by a religiously fervent and militarily skilled khaqan that greatly expands the Khazar Empire, blocking the coalescence of the Kievian Rus' in the process and rallying a number of other Turkic-Ugric-Finnic-Slavic-Whatever groups around his state. With this strengthened, religiously cohesive basis, his successors go into a conquering spree westward, like what the Magyars did IOTL but longer, with bigger numbers, religious fervor and political structures added to it.
The Frankish are devastated and a steppe-people elite of Jewish faith establishes itself as dominant over much of what used to be Charlemagne's Empire.
The rest is taken over by the Saracens, or the Byzantines, who temporarily profit of the Khazar destruction of the Bulgarian khaganate and the massive challenge they pose to the Caliphate.
The situation in the following centuries is confused, but gradually, while the steppe peoples confuse into the general population, Judaism sinks in as the dominant religion, and gradually spreads to the Norse as well, who make it even more widespread. Christians, however, remain a noticeable minority that enjoys the occasional support of the Byzantine Empire, until the Khazar successor states manage a sort of Fourth Crusade analog.
Paganism remains a larger force longer at least in the North, as Judaism isn't going a-crusading with the same enthusiasm, and Christian states (although ones regarded as heretical in Byzantium) survive in the British Isles and other peripheries. Muslims and Jewish powers clash with each other and occasionally try to pry the odd Christian country to their side or trample over it.
The process does not destroy peripheral Christianities such as Armenia and Georgia but weakens them significantly, to the point that by about 1600 Christians are hardly the local majority anywhere outside Ethiopia, the Scottish Highlands and Northern Mount Lebanon. However, there's still a sizable Christian minority everywhere west of India, with varying degrees of tolerance.
What happens to Spain is anyone's guess, but the devastations of the Khazar conquest are likely to weaken Europe and cut the Asturias' connection with the wider Christian world (I expect them to remain a Christian holdout for a time at least), so that nothing like Reconquista ever happens, if not much later.
The Mongols are butterflied, but Central Asia is likely to become another religious battleground where Judaism is much more stronger than IOTL, possibly shoring the spread of Islam in that direction.