Residents of South Yorkshire and Lincolnshire are being warned to remain vigilant as more rain is expected in previously flooded areas.

The Met Office issued several alerts for ice, rain and snow across the country on Thursday, while the Environment Agency warned further flooding could occur.

East Gloucestershire, parts of Wiltshire and south Wales were blanketed in snow on Thursday morning, while snow continued to fall in Bath as of 7am.

The EA said heavy rain was expected on Thursday, Friday and over the weekend, which could lead to further flooding for communities in South Yorkshire around the lower River Don.

Kate Marks, the agency’s flood duty manager, said: “Parts of Lincolnshire and the Midlands could also be affected by rain falling on already saturated ground over Thursday and Friday, as well as other parts of England as rain crosses the country from west to east.”

The former Conservative flood minister Richard Benyon told BBC Radio 4’s Today programme a flood “was an impossible time for anyone in authority to get it right” because “if you visit too soon, you’re in the way […] if you go too late, you seem somehow heartless, and you can never get that quite right”.

Benyon said he had walked through enough flooded houses to know why people were angry, but added: “They always say: ‘Where are the sandbags?’ Sandbags are in 99% of occasions useless. The other thing is ‘Why aren’t the rivers dredged?’ I remember someone in Somerset saying to me: ‘You know, all you have to do is dredge this river.’ And I said: ‘But the experts tell me that if we do that, the water will flow faster through your area and into people’s houses in Bridgwater.’”

When asked whether austerity had hampered the response to the floods, Benyon said the government had “spent more on flood defences than ever before”.

He said the flooding could potentially have an effect on the price and availability of certain specialist crops.

Lt Col John Holden from the 4th Infantry Brigade, who is leading the army’s cleanup services, told the Today programme Doncaster council and its partners had been working “24 hours a day from the start” and there was “a high state of readiness amongst all the agencies”.

Benyon said about 200 soldiers had been put on the ground in affected areas on Wednesday and accommodation and specialist engineer support had been provided. He said the army’s support would be available for as long as Doncaster council, the EA and their partners were asking for help.

On Wednesday, Boris Johnson was confronted by furious residents of Fishlake, who pleaded for help and accused the prime minister of not providing enough support when he visited the village near Doncaster.

Forty flood warnings remain in effect across the country, along with 75 flood alerts, which mean flooding is possible.

The EA said 300 staff were working around the clock in various parts of the country and pumps were being used in five of the worst-hit locations.

Since flooding began last Thursday, about 14,400 properties had been protected by flood defences, including 5,000 in South Yorkshire, the agency said.