Former foreign minister Alexander Downer, who recently replaced Mike Rann as Canberra's man in the UK, urged expats to vote Yes in any upcoming referendum.
"The first people of our country have for many years in our country been treated as second-class citizens," the high commissioner said at Australia House.
"We need as a country to raise their status to reflect the incredibly important role they played in the original formation of human civilisation in Australia."
Mr Downer said Aboriginal and Torres Strait Islander people, since European settlement, "had what you might, politely, call a hard time".
Changing the constitution would be a unifying moment, he said.
Australia House is the largest polling booth each federal election with around 15,000 expats usually casting a ballot there.
Prime Minister Tony Abbott in January promised to finalise by September a draft form of words for changing Australia's foundation document.
Any changes would have to be put to a referendum, which requires a majority of votes in a majority of states to be successful.
The Recognise campaign is raising awareness about the push to include indigenous people in the constitution and remove sections that discriminate on the basis of race.
Director Tanya Hosch was a guest at the high commission in London on Tuesday. She's also encouraging expats to support the campaign.
"There is a chance of failure here, so we have to actually act," Ms Hosch said.
"Despite that strong political and community support no one can take this for granted."
Ms Hosch said some people would seek, sadly, to derail the campaign.
"But we know there will be a great and terrible cost if we don't achieve this for our nation.
"The price we will pay is to condemn Australia, for another generation, to formal separation between indigenous and non-indigenous Australians."
Ms Hosch said constitutional recognition would enhance Australia's international reputation for fairness and justice.
British Prime Minister David Cameron's chief spin doctor, Australian Lynton Crosby, attended Tuesday night's event.
The former Liberal party strategist's research company, Crosby Textor, is advising the Recognise campaign.