The first of three French planes sent to evacuate European citizens took off from the capital, Niamey.
The coup has prompted demonstrations against the former colonial power, with the French embassy coming under attack.
However, France says it has no plans to repatriate about 1,000 French soldiers stationed there as part of efforts to counter Islamist militants.
Germany has urged its nationals to take up France’s offer to help other Europeans and Italy is also organising a flight.
It comes as the juntas in Burkina Faso and Mali warned that any forcible attempt to restore the ousted president would be seen as a declaration of war.
The two neighbours, also former French colonies, have both moved away from France and towards Russia, after staging their own coups in recent years.
Their warning marks a significant twist that could escalate the volatile situation in a region battling an Islamist militant insurgency.
Niger, which is rich in uranium, has been a key Western ally in the fight against jihadist extremism in the Sahel. Both France and the US have military bases there.
After Mali’s military leaders chose to partner up with the Russian Wagner mercenaries in 2021, France moved the centre of its regional counter-terror operations to Niger.
On Sunday, protesters outside the French embassy in Niamey chanted “Long live Russia”, “Long live Putin” and “Down with France”.
They also set fire to the walls of the embassy compound.
France sent planes because the closure of Niger’s airspace has made it impossible for people to leave by their own means.
There are an estimated 600 French nationals in Niger and fewer than 100 Germans.
Italy’s foreign ministry says there are about 90 Italians in Niamey out of a total of just under 500 countrywide, most of whom are in the military, the AFP news agency reports.
According to the Reuters news agency, Spain is also preparing to evacuate more than 70 Spaniards by air.
The UK is not organising an evacuation and has urged its nationals in Niger to stay indoors, while the European Union said it was not planning to remove its staff for the time being.
The US also said it was not evacuating its citizens, seeing no immediate threat to them or its facilities, AFP reports.
The situation in Niamey was reported to be calm on Tuesday.
One of the French evacuees, Anthony Garcia, said he was packing as little as possible for the journey: “We were asked to take just a small bag with essentials, because we can’t take the suitcase.”
Earlier, France had welcomed the ultimatum issued on Sunday by the West African bloc Ecowas, giving Niger’s junta a week to reinstate elected President Mohamed Bazoum, who has been confined to the presidential palace in Niamey.
Chad’s President Mahamat Idris Déby was in Niger the following day, leading mediation efforts on behalf of the Ecowas and was pictured with Mr Bazoum.
These diplomatic moves prompted Burkina Faso and Mali to issue a joint statement threatening that if Ecowas intervened militarily, they would withdraw from the bloc and go to the defence of their eastern neighbour.
They said such an intervention would be disastrous and destabilising.
Burkina Faso, Mali and Guinea are all currently suspended from Ecowas following coups in recent years.
Ecowas’s last major military intervention was in The Gambia in 2017, when Yahya Jammeh refused to step down as president after losing elections to Adama Barrow. After West African troops deployed, Mr Jammeh went into exile in Equatorial Guinea.
The regional body also sent troops to support Guinea-Bissau’s government last year following a failed coup attempt there.
Algeria, Niger’s neighbour to the north, has also urged caution over any possible military intervention, saying it “would lead to an escalation of the current crisis”.
Niger’s junta has not commented on the Ecowas demand, but vowed to defend the country from any “aggression” by regional or Western powers. It accused France of planning military intervention.
But on Monday evening, French Foreign Minister Catherine Colonna told French channel BFMTV the allegation was not true.
She suggested the coup, which has been welcomed by the shadowy Wagner group, could be an seen as an opportunity for Russia: “I am not sure that everyone in Niger sleeps with a Russian flag under their pillow.
“But it is possible that Russia tries to take advantage of the situation. It does it in other countries of the region. It’s an hypothesis.”
At a briefing on Tuesday, the White House’s National Security Council spokesman John Kirby said the US had no evidence that Russia was behind the coup.
According to Reuters, the evacuation plans will not affect operations of French nuclear fuel company Orano in Niger, as it said most of its staff were Nigerien nationals.