There are no invented religions in Harry Potter because that would make it a different kind of story. It's not high/epic fantasy like Lord of the Rings, set in an imaginary medieval-style world. It's set in a very real 20th-century world, just in a fictional esoteric subsociety within this real world. Because of this, the only religions mentioned or alluded to are those which exist in the real world.
Yes, there are references to religion in the Harry Potter books, albeit very few. The practise of appointing godparents is a Christian one, for one thing: so Sirius, Harry, James, Lily, Lupin, and Tonks were probably all at least culturally Christian. The quotes on the Potters' and Dumbledores' tombstones in Godric's Hollow are Biblical ones, confirming the Christian connection:
"For where your treasure is, there will your heart be also" (Matthew 6:21)
"The last enemy that shall be destroyed is death" (1 Corinthians 15:26)
The graveyard was a church one, which is even more evidence that the people buried there were Christians.
As for other religions, Anthony Goldstein has been confirmed to be Jewish, and Rowling has said she could imagine people of any and all religions being wizards and studying at Hogwarts - except Wiccans.
There are also a few references to God in the books, but these could be merely cultural and not reflect any religious adherence; many non-religious people say things like "thank God":
"Bill! Thank God, thank God –"
-- Molly Weasley, HP and the Deathly Hallows
"So that's little Scorpius. Make sure you beat him in every test, Rosie. Thank God you inherited your mother's brains."
-- Ron Weasley, HP and the Deathly Hallows