The voting age in Scotland is to be lowered to include 16- and 17-year-olds after David Cameron offered to speedily introduce new electoral powers for the Scottish parliament.

Despite opposing a change in voting age for UK elections, the prime minister bowed to pressure to give Scottish ministers powers to set a separate voting age and franchise for elections to Holyrood and for Scotland’s 32 local authorities.

Cameron agreed to try to fast-track the new powers through Westminster early in the new year after his first face-to-face talks in London with Scotland’s new first minister, Nicola Sturgeon.

The proposed change was one of the main recommendations of Lord Smith’s all-party commission on increasing devolution after the Scottish independence referendum, where some 124,000 16- and 17-year-olds were given the vote for the first time in a national poll.

Both sides confirmed there was a significant thawing of relations between the UK and Scottish governments, after more than two years of conflict during the referendum campaigns, which led to Alex Salmond’s resignation after the yes campaign’s defeat.

After her meeting with Cameron and earlier talks involving senior Cabinet ministers and leaders of the Welsh, Scottish and Northern Irish governments at Downing Street, Sturgeon said she was confident the new powers would be introduced.

“David Cameron and I are worlds apart in terms of political philosophy and outlook and our views on the constitution in Scotland, but yes, I think we can do business where we find common ground,” she said. “We are very confident that we will get the devolutionary power to extend the franchise to 16- and 17-year-olds in time for that to happen for the 2016 election. We need to move quickly on that. I think there is an understanding and a willingness to work with us.”

The official Tory policy is to keep the voting age at 18 in UK elections, despite growing numbers of senior Conservative backbenchers, including former minister Damian Green, pressing for a lower voting age across all elections.

But the Scottish Tories accepted the plan to allow Holyrood to control voting in devolved elections after being confronted by unanimous support for the change in the Smith commission talks from the SNP, Labour, the Lib Dems and Scottish Greens.

Downing Street said the talks between Cameron and Sturgeon, who took over from Alex Salmond as first minister last month, were “very cordial and constructive”. A spokesman said the prime minister made clear he wanted to forge “even stronger ties” between the two administrations and two parliaments.

Alistair Carmichael, the Lib Dem Scottish secretary, implied that Salmond’s resignation would make it far easier for the two governments to cooperate. “It takes two to re-set a relationship. I believe that Nicola Sturgeon will be much more constructive and co-operative to work with,” Carmichael said.

The new powers are to be passed to Holyrood in a section 30 order which will amend the Scotland Act rather than introduce wholly new legislation. Sturgeon is pressing for that to be confirmed by a privy council meeting in March, the last before the 2015 general election.