The Edinburgh celebrations, a major tourist draw which last year attracted some 30,000 people, had been due to feature a fireworks display and a concert headlined by Scottish band Texas.
But organisers said it would not be safe to go ahead with preparations for outdoor events scheduled for Monday and New Year's Eve on Tuesday due to "ongoing high winds and inclement weather".
Planned New Year fireworks in the holiday resort town of Blackpool in northwestern England were also cancelled due to a forecast of high winds, the local council said.
The Scottish Environment Protection Agency (Sepa) issued a rare severe flood warning of "danger to life" for parts of Scotland.
It said a heavy downpour was expected to burst riverbanks in Aviemore in northern Scotland and nearby areas of the Scottish Highlands.
"Due to persistent and heavy rainfall, river levels on the River Spey will rise throughout Monday night and into Tuesday morning causing serious flooding," it said.
"Extensive flooding to properties and businesses is expected in and around Aviemore."
The UK's Met Office, meanwhile, said heavy rain and strong winds could be expected in parts of Scotland, Northern Ireland and northern England.
Some snow was also forecast for parts of Scotland.
"A series of low-pressure systems will track across the UK over the next couple of days bringing some potentially disruptive weather," said Met Office meteorologist Steve Willington.
"Almost the entire UK is covered by at least one weather warning" over the coming week, he added.
A Met Office amber warning for rain and snow -- the second most severe weather notice, meaning there is a potential risk to life -- is in place for Scotland.
A number of less severe yellow warnings for rain, wind and snow were also in place for Scotland and northern England.
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