Saturday will see a brief return to sunnier conditions, before thundery showers set in, the Met Office said.
Travellers and holidaymakers faced delays on wet roads in areas on Friday, and downpours brought services at several London Tube stations to a halt.
A Met Office warning for wind and rain remains in place for south-east England until late morning.
Cloud is then expected to break providing drier, brighter conditions.
Temperatures could reach 21C (70F) in London, the high teens in the Midlands, and around 15C (59F) in Scotland and Northern Ireland.
Forecaster Alex Burkill said: "Most places should be dry by lunchtime today.
"It should be a much drier, brighter and much more pleasant day."
Sunday will see conditions take a turn for the worse, although it is not expected to be as wet as Friday, Mr Burkill added.
"By the time most people wake up on Sunday morning it is going to be a rather cloudy day."
Most areas of the UK are expected to see between five and 10mm of rainfall, except for Northern Ireland, which is predicted to see as much as 30mm.
Rain is expected to move eastwards throughout the day, with blustery and wet conditions set to last into early next week.
"Monday is looking pretty blustery with some heavy, thundery showers expected, but those should ease by the middle of the week," Mr Burkill said.
Heavy rain is due to return on Sunday after a month's worth of rain fell in a single day on Friday, forecasters say.