Hundreds of civilians are believed to have been killed in South Ossetia
The Foreign Office is advising British nationals in Georgia to leave unless there is an urgent need to stay.
Officials earlier advised against all non-essential travel to the conflict-hit country, but stepped up their warning after further fighting.
Violence has erupted between Russian and Georgian forces over control of the breakaway South Ossetia region.
Gordon Brown held talks with international leaders on Sunday in a bid to create pressure for a ceasefire.
A Downing Street spokesman said the UK prime minister had "detailed discussions" with the French President Nicolas Sarkozy and had also spoken to UN Secretary General Ban Ki-moon.
He added: "The foreign secretary has been in touch with US Secretary of State Condoleezza Rice and EU foreign ministers.
"We will continue to press for international action to help bring the conflict to an end and we urge both parties to agree an immediate ceasefire."
On Friday, British Ambassador to Georgia Denis Keefe wrote to British citizens in the country advising them to stock up on water, food and fuel.
He said then that there was "no immediate cause for alarm" and British nationals were not being advised to leave the country.
However, on Sunday the Foreign Office issued new advice.
It said in a statement: "If you or your family have no urgent need to remain in Georgia you should leave as soon as possible.
"It is wise to do so while some air services are still available and the border remains open."
The Foreign Office is also advising against all travel to the separatist regions of South Ossetia, Abkhazia and the upper Kodori.
It said serious fighting was continuing in South Ossetia and there had been aerial bombardment of Georgian military installations outside the conflict zone at the Black Sea port of Poti and the military bases at Vaziani, Senaki and Marneuli.
British charity worker Sian Davis, from Reading, told the BBC the mood in the Georgian capital Tbilisi was subdued.
She said: "Tbilisi looks fairly normal. It's pretty calm, but it's really, really quiet, eerily quiet, and last night the streets were empty, the restaurants were empty.
"Everyone was either at home or had packed up and moved out of the city.
"People are really, really scared. Even on the bus, there's people on the bus, but no-one's speaking to each other, no-one's joking, no-one's laughing and this is not Georgian normality, because Georgian normality is noisy and crazy and people laughing and joking.
"This is the problem - people are panicking."
Miss Davies, who is taking part in a European Voluntary Service project in Georgia, said she knew of Britons who were already fleeing the country.
"They are advising us to go and get to the Armenian border," she said. "A couple of my British friends have done that already.
"A few of us are still here, but should the situation become a lot worse very quickly, we will have to leave as well."
Mr Miliband has said the government is "deeply concerned" by the violence in Georgia.
He has held talks with European foreign ministers and with US Secretary of State Condoleezza Rice over how to respond to the crisis.
Based on Russian and South Ossetian estimates, the death toll on the South Ossetian side is believed to be at least 1,500, mostly civilians. Georgian casualty figures range from 82 dead, including 37 civilians, to a figure of about 130 dead.