The storm, unusual for August, has also led to the flood alerts on rivers in Shropshire after heavy rain fell overnight in Wales.
The Environment Agency has posted flood alerts for the Severn Vyrnwy Confluence and for the Upper Teme.
In Bucknell, affected by flooding during Storm Ellen, there are reports that a shop window has been smashed by the force of water thrown up from vehicles drive through the floods, while in Wem a tree has come down blocking Lowe Hill Road.
Elsewhere, homes have been flooded, rail lines blocked and campers rescued as Storm Francis begins to lash the UK with strong winds and heavy rain.
The incidents came as the Met Office issued an amber warning for very strong winds across most of Wales and central England between 2pm and 10pm on Tuesday.
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Gusts of up to 65mph could be felt inland, with forecasters warning of disruption to transport, power cuts and potential flying debris that could lead to “injuries or danger to life”.
Emergency services have already warned the public to take extra care in the stormy conditions across the UK, particularly along the coast.
South Wales Police said they were involved in two separate water searches from the swollen River Taff on Tuesday.
A spokeswoman said emergency services were searching the River Taff in Cardiff following reports of a person having entered the water near the Principality Stadium shortly before 8.40am, while a river search was under way in the river in Taff’s Well following reports of a canoeist having capsized.
A woman was also rescued at the River Ely in Leckwith following reports of a person in difficulty, the force said on Twitter.
Meanwhile, fire crews had to rescue holidaymakers from a flooded campsite in the town of St Clears, Carmarthenshire, after river levels rose in the area.
Mid and West Wales Fire and Rescue Service said nine people and two dogs were rescued by fire service personnel using a swift rescue sledge, lines and wading gear.
A number of homes in Wales were also said to have been hit by flooding in Llanelli, Neath, Whitland and Tonyrefail, while some roads across the country were left underwater.
Elsewhere, travellers were warned of flooding disrupting rail services and trees blocking roads on Tuesday morning.
According to the Met Office, gusts of 67mph were recorded at the Isles of Scilly between 8am and 9am on Tuesday morning, while they reached 73mph at the Needles on the Isle of Wight in the same period.
An Environment Agency recording taken between 11pm on Monday and 7am on Tuesday logged 65.8mm dropping at White Barrow in Devon.
In the same period, Natural Resources Wales recorded 61.4mm falling at Tavernspite in Carmarthenshire, Wales, while Spite in Glamorganshire saw 56.2mm.
Three Met Office yellow weather warnings of heavy rain or strong winds cover most of the UK on Tuesday, with stormy conditions expected to last until Wednesday morning.
Warnings of rain cover Northern Ireland, southern Scotland, northern England and parts of North Wales.
Forecasters predict gusts of up to 70mph could batter exposed coasts and hills across Wales and most of England.
The M48 bridge across the River Severn has been closed in both directions due to the increased wind speeds.
The Highways Agency reported a fallen tree temporarily blocking the A30 in Cornwall, while another toppled tree brought disruption by blocking the rail line between Gunnislake in Cornwall and Plymouth in Devon, before being cleared.
At Neath in South Wales, flooding caused lines to be closed for part of the morning, with knock-on delays of up to 60 minutes.
Flooding on the line between Fernhill and Aberdare, also in South Wales, triggered a suspension to services.
In Northern Ireland, police reported an incident of a river bursting its banks near Newcastle, as well as roads blocked by flooding, a fallen tree and downed power line.
As of midday on Tuesday, the Environment Agency has issued 22 flood alerts for England, largely in the South West and West Midlands.
Natural Resources Wales had put out three flood warnings – advising immediate action – and 17 flood alerts for rivers across the south west.
The Met Office has never had two named storms in August since the process started in 2015, but Francis comes on the back of Ellen, which struck last week and caused power outages.
Storm Ellen also saw 15-year-old Nicola Williams swept to her death in the Rhymney River in Llanrumney, Cardiff, and a 50-year-old holidaymaker die in the sea near Helston, west Cornwall, after getting into difficulties.