South Wales, where a major incident has been declared by one council, has seen over 100mm of rain in places - in line with suggestions that some areas could see a whole month's rain over the weekend.
In Pontypridd, residents have been seen bailing out water from outside their homes, while cars are submerged after a river burst its banks.
Forecasters say the worst of the rain is now moving slowly eastwards into England, where the midlands and south west could also see flooding. More than a hundred flood warnings are in place across the UK.
Yellow warnings for wind and rain remain in effect for western Scotland, southern England and Wales, as well as Northern Ireland on Sunday.
There are also hundreds of flood alerts in place in England, Wales and Scotland.
Wind gusts have reached 75mph in coastal areas and up to 65mph inland.
Around London, all of the Royal Parks are closed on Sunday due to high winds - including Hyde Park and its popular Winter Wonderland attraction.
The parks' management said there would be delayed reopenings on Monday following a safety inspection.
On Saturday, roads, railway lines and flights at some airports were impacted due to floodwater, high winds and fallen trees. At least one person has died as a result of the storm.
More than 24,000 properties were also without power across the UK on Sunday afternoon, according to PowerOutage, a data aggregator.
In south Wales, a major incident has been declared by Rhondda Cynon Taf Council after significant flooding. The River Taff has burst its banks in Pontypridd, flooding nearby homes.
Three rest centres have been set up for affected residents, the council leader said.
In Denbighshire, north Wales, five adults and five children were rescued from a house flooded with debris following a landslide on Saturday.
Following hours of heavy rain caused by Storm Bert, residents in Dundonald in Northern Ireland were stranded in their homes due to the floodwater on Saturday. Up to 10 homes may have also been flooded in Coalisland.
Heavy rain will gradually move eastwards away from Wales and south-west England and linger for much of Sunday afternoon across parts of Yorkshire and the Midlands, as well as east and south-east England.
Behind this front, conditions are expected to brighten with a few blustery showers.
In Scotland and Northern Ireland, there will be a mix of sunny spells and squally showers, particularly areas closer to the centre of the storm in the far north-west.
Despite the wind and rain, it will be an incredibly mild day with highs of 9-15C. Parts of south-east England could see temperatures peak at 17-18C.
It comes after a cold snap in the UK for much of the last week that saw snowfall in several regions and temperatures drop below freezing.
West Midlands Fire Service said that a person had been rescued from a car stranded in floodwater in Walsall on Sunday morning.
Since Storm Bert made landfall on Friday, three men have died on the roads.
A man in his 60s died on Saturday after a tree hit his car near Winchester, and two others died in crashes - one in West Yorkshire and another in Northamptonshire - though the latter incidents have not been directly attributed to the storm.
Parts of Scotland are experiencing a second day of disruption with high winds and localised flooding after Storm Bert swept heavy snow and rain across the country.
Milder temperatures are causing the snow which covered the north of England and much of Scotland earlier this week to melt.
Amber weather warnings for snow were in place on Saturday for parts of Scotland and north-east England, while large swathes of the UK were under yellow weather warnings.
