The local area is on the edge of a large zone covered by the lowest-level yellow alert, which includes the west of England and Wales.
The warning is from 3am tomorrow morning until about midnight, and the Met Office predicted rainfall of up to 50mm within three hours in some places.
There is a “small chance” that homes and businesses could be flooded quickly, it added, with damage to some buildings from floodwater, lightning strikes, hail or strong winds.
Transport could be affected by flooding or lightning strikes, with the potential for difficult driving conditions, road closures and power cuts.
The weather will break the warm conditions that have bathed the area in recent days, giving a boost to the local tourist industry.
The warning said: “Thunderstorms and outbreaks of heavy rain will arrive across the far south-west early on Tuesday and then spread further north-east across England and Wales whilst also increasing in frequency during the day.
“In a few places there is the potential for 30mm or so of rain to fall in an hour or two and 30-50mm of rain to fall in three hours.
“Parts of northern and eastern Wales, the western Midlands and southern parts of northwest England are thought most likely to see the highest rainfall totals.”
A similar weather warning was also issued for Wednesday, predicting thunderstorms in the Midlands, northern England, and Scotland.