Australian prime minister Malcolm Turnbull has said his country wants to seal a free trade agreement with the UK "as soon as possible" after Britain's withdrawal from the European Union.
Mr Turnbull's comments came after he held talks at 10 Downing Street with Theresa May, who revealed that International Trade Secretary Liam Fox will visit Australia in the "coming months" as part of ongoing talks on an agreement.
The two prime ministers visited the scene of last month's London Bridge terror attack to thank emergency workers who went to the aid of victims, who included two Australian nationals.
Mr Turnbull's first official visit to the UK as prime minister will also include an audience with the Queen.
Speaking alongside Mrs May in Downing Street, the Australian PM said: "As Britain moves to completing its exit from the EU, we stand ready to enter into a free trade agreement with the UK as soon as the UK is able to do so.
"Once Brexit is achieved, we look forward to speedily concluding a free trade agreement. At the same time, we are looking forward to the early conclusion of a free trade agreement with the EU."
Mr Turnbull said he hoped the EU deal could be finalised before the expected date of Brexit in March 2019.
And asked how quickly a UK agreement could follow, he said: "As soon as possible. We move quickly. Australians are fleet of foot, we don't muck around, we are very simple. So we will move as quickly as the UK is able to move."
Mr Turnbull said he recognised Mrs May's vision of Brexit as a chance for Britain to take a new global role with "big horizons and big opportunities" and said he wanted Australia to be a partner in that drive.
"There are no two nations in the world that trust each other more than the UK and Australia," he said. "We are family in a historical sense, we are family in a genetical sense. We are so close, and that trust is getting stronger all the time."
Voicing Australia's solidarity in the wake of the wave of terror attacks on the UK, Mr Turnbull said: "When your institutions are attacked, we feel that ours are too. When Britain is attacked by terrorism, we feel we are attacked as well.
"The attack on Westminster Bridge, the cruel assault on young children in Manchester Arena, the attack at London Bridge and Borough Market - these were felt by the Australian people as keenly as the attacks we have suffered at home in Australia.
"We stand shoulder to shoulder now, as we always have, in freedom's cause today, defying and defeating the Islamist terrorists that seek to do us harm, to seek to destroy our way of life and prevent us from living our way of life as we always have."
Mrs May said that she and Mr Turnbull had discussed co-operation in the fight against terrorism and in denying the use of the internet as a "safe space" for extremist groups.
On their visit to Borough Market with Metropolitan Police chief Cressida Dick, the two premiers met police officers who attempted to revive Australian Sara Zelenak, who was among the eight people killed.
They went to the spot where Australian nurse Kirsty Boden, who rushed to the aid of those attacked, was herself murdered by the three attackers.
Mr Turnbull said: "It was very moving to meet them. The three of us did our best not to burst into tears."
He added: "I want to say how much we admire the outstanding response of the police, arriving on the scene so quickly, dealing with the terrorists so effectively and decisively.
"They ran towards the danger while others were fleeing the danger.
"We say to these killers, to these terrorists that seek to change the way we live, we will not be cowed. We won't change the way we live. We will defy you and defeat you."
Mrs May praised Ms Boden, saying: "We should thank her and the others who ... didn't think of themselves but went to help those who they saw in need."
She told Mr Turnbull: "The resilience of Borough Market is fantastic. We are going to stand up against these terrorists and we are, as you say, going to defeat them. They will not destroy our way of life, and our values will prevail."
Mrs May said a trade deal with Australia was a "priority" for the UK after Brexit, to expand on the £14 billion-worth of trade between the two nations.
"We've both made clear our intention to continue to deepen our trade and investment relationship as the UK leaves the EU," she said.
"Our Brexit negotiations have started well, and I have made clear to prime minister Turnbull that an ambitious and comprehensive bilateral trade deal with Australia remains a priority for the UK.
"Australia was the first country with whom the UK established a trade working group following the vote to leave the EU and we're keeping up a regular and productive dialogue on the future of our free trading relationship."
Mr Turnbull joked about his time as a contemporary of Mrs May and her husband Philip May at Oxford University.
"We're not really debating today but the last time we debated was at the Oxford Union - Theresa Brasier and Malcolm Turnbull were both on the notice paper and the president was Philip May," he said.
"So now what we need is president May here to oversee the debate."
And Mrs May gently ribbed her old friend about England's first cricket Women's World Cup victory over Australia since 1993.
"It's always a pleasure to welcome our Australian friends to London, and even more so when you've just beaten them at the cricket," she said.
Mr Turnbull stressed that the UK and Australia remained close friends "whatever our differences may be on particular sporting contests", adding: "Australians want their team on the field beating the Poms."
Outside Paul Wheeler Fresh Supplies in Borough Market - the stand by which the attackers were shot - owner Paul Wheeler spoke to both prime ministers about how it has been since the incident and the support they have had.
The 51-year-old told the Press Association: "The Australian PM seems all right, I told him he had brought the Australian weather with him, and I thanked the Commissioner, just to say about the boys who were here on the night and after."
Federico Fugazza, of Argentinian food stand Portena, told them how his staff barricaded themselves in behind a metal shutter, and watched the attack unfold.
The 41-year-old said: "They were really exposed but somehow they stayed safe, and there was a customer inside - we were lucky.
"The prime ministers were thankful of that, and in a way proud of how most people reacted and how everyone tried to stay safe and together."
Labour MP Ben Bradshaw, a leading supporter of the Open Britain group, said: "A free trade deal with Australia would be good news, but with it accounting for just 1.4% of our exports, it is a drop in the ocean compared to the EU market on our doorstep which accounts for half our trade.
"This demonstrates the foolishness of betting the house on trade deals which may not happen for years, when the Government is committed to quickly pulling us out of the single market and the Customs Union, a combination of which gives us free trade across the European Union and trade deals with over 50 other countries."