Had Islam risen as a separate branch of Christianity .what would the World look , what would be their relationship with Europe and the conquered people.
The question is what kind of Christianity is it.
I am assuming that the following points, but let me know if that is not what you mean.
1) "Islam" accepts Jesus not just as a prophet of God, but actually his "son".
2) "Islam" rejects the Trinity so Jesus is not truly divine, but still mortal. It is similar to both the Arian heresy and Unitarianism in that respect.
3) God made Mohammed as his prophet in order to "correct" the heresy of existing Christian denominations and return to the "true" message Jesus intended.
There are several consequences.
A) Muslims may not be able to count on the help from Jews which they had very early on since their adherence to Jesus is problematic.
B) Assuming the Conquests happen as IOTL, they'll have a much more difficult time. Instead of seeing Muslims as a third party that will allow them to worship as they wish, the Coptic Church will be more aghast at the Muslim invaders who bringing yet another heresy to their lands, and they will be less likely to leave the Coptics in peace.
I think this will mean the Muslims will have a much harder time expanding their empire as they encounter more resistance. "Islam" may not expand west of Egypt as it becomes bogged down in debates and schisms in its attempts to rule Christians of other schools.
It will also mean they'll have a harder time resisting the Byzantine resurgence since they already accept some Christian doctrine and its members will be more susceptible to conversion to the more orthodox school.
It also means that much of Persia and Central Asia becomes Christian, whether ultimately it is the Muslim variant or the native Nestorian Church.
This Muslim Christianity survives as yet another non-Chalcedonian version in the Middle East which also spreads to East Africa and India, but I don't see it as spreading as much Islam did IOTL. This will have a big impact on how Europe and the Latin Church develops.