More than 800 properties across parts of England have been affected by flooding so far, with the Met Office warning of more rain on the way.
The Environment Agency said further wet weather over the next 48 hours could bring “severe” flooding to areas already affected by rising waters.
Forty-five flood warnings are in place as of midday on Thursday across England, as well as 122 flood alerts.
Oxfordshire County Council said firefighters have rescued “a number of people” from vehicles stuck in floodwater.
In a tweet showing cars trapped on a flooded road in Woodeaton, north-east of Oxford, the council urged drivers to avoid flooded routes.
Meanwhile, images posted online show the Oxfordshire village of Wendlebury, near Bicester, flooded after the nearby Wendlebury brook burst its banks due to heavy rain.
Wendlebury Parish Council advised people living near the brook to move their belongings “upstairs if at all possible”.
In Worcestershire, West Mercia Police said the A448 Kidderminster Road in Bromsgrove was “completely impassable” near Fox Lane and through Chaddesley Corbett.
The Met Office warned there could be a danger to life in parts of the East Midlands and Yorkshire as more rain is forecast to fall on the already flooded region on Thursday.
It has issued a yellow weather alert for areas including Nottingham, Lincolnshire, South Yorkshire and Derby, warning of “heavy and persistent” rain until 10pm on Thursday.
A yellow weather alert for “widespread” rain is also in place for parts of Wales, south-west England and the West Midlands until 11.45pm.
The Environment Agency said since flooding began last Thursday, around 14,400 properties had been protected by flood defences, including 5,000 in South Yorkshire.
More than 800 properties have been affected by the floods, with people evacuated from homes in Bentley and Fishlake, near Doncaster, the agency said.
Around 38 pumps are on site in South Yorkshire to remove 50 million litres of water per hour to protect homes and businesses, the agency added.
Pumps set up around the village of Fishlake have removed 1.25 million tonnes of water in the past 24 hours, reducing water levels from two metres to 0.3 metres, the agency said.
The agency has urged communities to check for flood warnings and follow the advice of emergency services.
It said about 200 operational staff have been on sites in Yorkshire assisting with pumping and deploying sandbags ahead of further rain later on Thursday.
Kate Marks, the Environment Agency’s flood duty manager, said: “Heavy rain today is expected to cause river and surface water flooding across parts of England, continuing into tomorrow in some areas, as well as potentially contributing to ongoing flood impacts in Yorkshire, Derbyshire, Nottinghamshire and Lincolnshire.
“Today we ask them to please remain vigilant and take steps to prepare for flooding by checking their flood risk regularly and making plans to stay safe.”