Taoiseach (Irish Prime Minister) Micheál Martin has wished the UK government "well in the challenging times ahead".
He said the UK was in a "time of political change and some uncertainty" as it prepared to appoint a new prime minister – its third within two months.
On Monday, it was confirmed Rishi Sunak would succeed Liz Truss in Number 10.
"A stable and prosperous Britain is in all our interests," Mr Martin said.
He was addressing UK and Irish politicians at a meeting of the British-Irish Parliamentary Assembly.
Sinn Féin leader Mary Lou McDonald said the new prime minister will face a challenge of ensuring Northern Ireland "does not remain endlessly in limbo" without government.
Speaking in Belfast, she accused the DUP of "inertia and inaction, and do-nothing politics".
Ms McDonald said Downing Street should quickly demonstrate a willingness to get Stormont restored.
"There certainly can't be any further delay," Ms McDonald said.
"We need to see an end to the brinkmanship, to the bad faith and we need to see now a good faith engagement and the making of a deal, which by the way and we have said this repeatedly, can be done very, very quickly."
In Mr Martin's speech on Monday morning, he called for better cooperation between the UK and Irish governments on the issue of the Northern Ireland Protocol.
The protocol – the post-Brexit trade arrangement for Northern Ireland – has been a source of tension since it came into force at the start of 2021.
Talks between the UK and the EU have resumed in an attempt to resolve the issue but there is no sign of an imminent deal.
"To the incoming British prime minister I want to stress the importance of the two governments working in partnership to support the gains of the Good Friday Agreement," said the taoiseach.
"I urge the new British prime minister to move quickly to genuine and substantive engagement with European Union."
The British-Irish Parliamentary Assembly – made up of politicians from both jurisdictions – meets twice a year, with this event focussing on trade and the economic effect of Brexit.
Those attending the event also heard from Northern Ireland Office Minister Steve Baker.
He said that regardless of who became the UK's new prime minister, the government would stick to its existing policy on the Northern Ireland Protocol.
The devolved government at Stormont has not functioned fully since February when the Democratic Unionist Party (DUP) withdrew from the executive in protest at the protocol.
The DUP believes the protocol undermines Northern Ireland's place within the UK by effectively creating a trade border in the Irish Sea.
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