(Adds NATO, Georgian quotes, attack on politician)
* Crackdown will be in stages, president says
* Ukrainian troops and APCs seen near flashpoint town
* U.N. report says threat to Russian speakers exaggerated
* German firm starts sending gas to Ukraine
* Poland calls for more NATO reinforcements
KIEV/SLAVIANSK, Ukraine, April 15 (Reuters) - Russia declared Ukraine on the brink of civil war on Tuesday as Kiev said an "anti-terrorist operation" against pro-Moscow separatists was under way, with troops and armoured personnel carriers seen near a flashpoint eastern town.
Twenty-four hours after an Ukrainian ultimatum expired for the rebels to lay down their arms, witnesses however saw no signs yet that Kiev forces were about to storm state buildings in the Russian-speaking east that armed militants have occupied.
Interim President Oleksander Turchinov insisted the operation had started in the eastern Donetsk region, although it would happen in stages and "in a considered way".
Amidst the deepest East-West crisis since the Cold War, the leaders of Russia and the United States have called on each other to do all in their power to avoid further bloodshed.
The standoff has raised fears that Moscow might turn off gas supplies to Kiev, disrupting flows to the European Union. Russian exporter Gazprom promised it would remain a reliable supplier to the EU, but German energy company RWE began deliveries to Ukraine on Tuesday - reversing the usual east-west flow in one central European pipeline.
Russian Prime Minister Dmitry Medvedev gave a gloomy assessment after at least two people died on Sunday when Kiev unsuccessfully tried to regain control in the town of Slaviansk, about 150 km (90 miles) from the Russian border.
"Blood has once again been spilt in Ukraine. The country is on the brink of civil war," he said on his Facebook page.
Turchinov said the offensive, which he first announced on Sunday, was finally underway. "The anti-terrorist operation began during the night in the north of Donetsk region. But it will take place in stages, responsibly, in a considered way. I once again stress: the aim of these operations is to defend the citizens of Ukraine," he told parliament.
At least 15 armoured personnel carriers displaying Ukrainian flags were parked by the side of a road around 50 km (30 miles) north of Slaviansk, witnesses said.
Ukrainian troops wearing camouflage gear and armed with automatic weapons and grenade launchers were stationed nearby, with a helicopter and several buses containing interior ministry personnel near the road.
ALL QUIET IN SLAVIANSK
In Slaviansk itself, where separatists have seized the local headquarters of the police and state security service, a Reuters correspondent heard no shots or blasts.
Outside the occupied police station about a dozen civilians manned barricades that have been built up overnight with more tyres and wooden crates. A dozen or so armed Cossacks - paramilitary fighters descended from Tsarist-era patrolmen - stood guard at the mayor's offices. Shops were functioning as usual and bread supplies were normal.
"The night passed quickly, thank God. There have been lots of rumours of violence, but it's been very quiet here. We are in control," said one civilian on the barricades outside the police station, who gave his name only as Rustam.
Igor Dyomin, a spokesman for the Donetsk region police, said militants had ended their occupation of the police station in Kramatorsk, which had begun on Saturday. "They left of their own accord," he said. But later a state security service spokeswoman said rebels had now occupied the agency's local offices in the same town.
In Kiev, a radical pro-Russian candidate running for Ukrainian presidential elections due next month was beaten up by an angry crowd.
GREATLY EXAGGERATED STORIES
Ukraine's security forces have been in some disarray since protesters ousted pro-Moscow President Viktor Yanukovich in February. However, the interim leadership may also be treading carefully with the offensive to avoid civilian casualties.
Moscow accuses Kiev of provoking the crisis by ignoring the rights of citizens who use Russian as their first language, and has promised to protect them from attack. It also highlights the presence of far-right nationalists among Kiev's new rulers.
However, a United Nations report on Tuesday cast doubt on whether Russian-speakers were seriously threatened, including those in Crimea who voted to join Russia after Moscow forces had already seized control of the Black Sea peninsula.
"Although there were some attacks against the ethnic Russian community, these were neither systematic nor widespread," said the report by the U.N. human rights office.
The report cited "misinformed reports" and "greatly exaggerated stories of harassment of ethnic Russians by Ukrainian nationalist extremists".
NATO Secretary-General Anders Fogh Rasmussen accused Moscow of involvement in the rebellions. "We never ... comment on intelligence, but I think from what is visible, it is very clear that Russia's hand is deeply engaged in this," he told reporters at a meeting with EU defence ministers in Luxembourg.
Russian Foreign Minister Sergei Lavrov denied that Moscow was stirring up the separatists in the east and southeast as a possible prelude to repeating its annexation of Crimea. "Ukraine is spreading lies that Russia is behind the actions in the southeast," Lavrov said on a visit to China.
He called on Kiev to hold back before a meeting between Russia, the EU, United States, and Ukraine planned for Geneva on Thursday. "The use of force would sabotage the opportunity offered by the four-party negotiations in Geneva," he said.
Moscow has demanded constitutional change in Ukraine to give more powers to Russian-speaking areas, where most of the country's heavy industry lies, while the secessionists have demanded Crimean-style referendums in their regions.
Kiev opposes anything that might lead to the dismemberment of the country. But in an attempt to undercut the rebels' demands, Turchinov has held out the prospect of a nationwide referendum on the future shape of the Ukrainian state.
RWE's deliveries of gas to Ukraine through a pipeline from Poland marked an initial step in EU efforts to counter the risk that Russia will turn off the taps.
Central Europe's pipeline network is designed to carry Russian gas westwards. But Polish operator Gaz-System said it had reversed the flow to send back 4 million cubic metres per day, the equivalent of 1.5 billion annually - a modest volume compared with Ukraine's need for more than 50 billion.
Moscow has nearly doubled the price it charges Kiev this year, and President Vladimir Putin has threatened to halt supplies if Kiev does not repay more than $2 billion it owes to Gazprom. Putin has also warned EU leaders that this could disrupt their supplies that flow across Ukraine.
Despite the bad east-west climate, energy giant BP said its Russian business was unaffected by sanctions the United States and EU have imposed on people close to Putin. BP owns a 19.75 percent stake in Kremlin-controlled Rosneft, which became the world's top listed oil producer last year.
"We are rock solid with our investments in Rosneft and (we) will stand by our investments. For us it's business as usual," chief executive Bob Dudley told reporters in Moscow.
NATO states have temporarily sent troops, aircraft and ships to eastern Europe to reassure nervous post-communist alliance members. But Polish Defence minister Tomasz Siemoniak said the Crimean crisis made it vital that NATO station significant numbers of troops there and ignore any objections from Moscow.
"What is really important is the strengthening of NATO's eastern flank," Siemoniak told Reuters.
The president of Georgia, which fought a brief war with Russia in 2008, also expressed concern.
"Georgia, which experienced very harsh Russian foreign policy moves six years ago, and which is still in a very complicated relationship with the Russian Federation, is naturally very alarmed because of the Ukrainian precedent," President Georgy Margvelashvili told Reuters.
(Additional reporting by Serhiy Karazy and Conor Humphries in Kiev; Alessandra Prentice and Alexei Anishchuk in Moscow; Ben Blanchard in Beijing; Margarita Antidze in Tbilisi; Adrian Croft in Luxembourg; Tom Miles and Stephanie Nebehay in Geneva; Marcin Goettig and Marcin Goclowski in Warsaw; Writing by David Stamp; Editing by Will Waterman and Giles Elgood)