It depends on what you mean by "modern religion". The major religions in the modern world would be Judaism, Hinduism, Christianity, Islam, and some forms of Buddhism
They all began between around 4,000 (Judaism) and 1,400 (Islam) years ago.
However, all (with the exception of original Buddhism) have changed a great deal down the years in terms of what is considered moral, and what is considered to be true.
Thus, if you mean: "when did the modern *versions* of religion arise", then the answer would simply be that the religions change to suit the morality of the society. So the modern versions arose when modern morality arose. Religions generally take longer to change than society does, though.
Since morality changes gradually, it would be very difficult to say for sure, but you could take some landmark events - the abolition of the death penalty in Europe, or women getting the vote and being regarded as equal, for example.
Having said that, there are still many people in the modern world who take their morality from the a skewed form of the older forms of religion. Some people believe that homosexuals should be imprisoned, or that women should stay at home, or that teenagers shouldn't learn about sex. These people, with their outdated morals, manage to do a lot of harm to modern society. Answer: There are many "modern" religions. Not even counting sects slitting off from the old religions these would include Scientology. Rastafarianism, Wicca, Jedi , Church of the Subgenius etc. The newest of them are probably coming into existence just now. As an example, in many jurisdictions all that is required to start an "official" religion (just as good and any that ever have been in the eyes of the Law) is to fill out a registration form and fork over the fees and promise to comply with the annual reporting criteria.
Some shintô rituals present since paleolitic, but this religion became this modern form only since 18th centery. Just for me, shintô is the oldest religion of the modern world with the exeption shamanism, but this religion evolutions. Sorry my terrible English.
In modern day Israel, although many of the early churches were founded across modern day Turkey.
It depends on the religion.
THE modern history of Jehovah's Witnesses began began in Allegheny, Pennsylvania, U.S.A., which is now a part of Pittsburgh.
It is the native religion of Japan and was once its state religion.
Judaism did not begin within another religion.
India
About 550BCE
Islam
Polytheists.
No religion is called old or modern religion. God religions are revealed by God and remains as assigned and commanded by God. The main religion of Persians is Islam.
It began in India. This religion is the oldest religion in the world.