The country's electoral commission cancelled them last week, just as the polls were about to open.
The Independent National Electoral Commission (INEC) blamed logistical problems for the delay, denying political pressure.
In a statement titled "Welcome to a New Nigeria", the council's spokesman for Africa, Bishop Seun Adeoye said: "A lot of issues have been raised about Nigeria's general election and the day is here.
"When two or more people contest in an election, only one person will definitely be declared as the winner. This should be the mindset of all politicians.
"[We] therefore urge politicians to accept the outcome of the results of the election in the spirit of sportsmanship. We should not drag the country to unnecessary crisis. War is costly. Nigeria must not go to war.
"We also call on President Muhammad Buhari to build on the legacy built by former President Goodluck Jonathan who we are proud of and who showed a trait of a good Christian by accepting the results of the 2015 presidential election."
The winner is likely to be either Nigerian President Muhammadu Buhari or opposition challenger Atiku Abubakar. Both are Muslim with Christian running-mates.
President Muhammadu Buhari (below) has cast his ballot as he seeks a second term in a race that observers now say is too close to call with top challenger and former vice president Atiku Abubakar.
Andrew Boyd from Release International told Premier: "In the first six months of 2018 [in Nigeria] 6,000 people were killed and 50,000 Christians were driven from their homes.
Meanwhile, gunfire has been heard in at least two cities shortly before the polls opened, but police in Maiduguri in the north east called the blasts there a show of force by security forces.
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