The winds are expected to ease in South Yorkshire this morning, but a weather warning for strong wind remains in place until around 10am today.
The storm hit the country overnight, bringing gusts of up to 75mph.
Highways England has urged drivers to take extra care on their journeys to work, warning that people should delay travelling if the weather becomes too severe.
Those driving caravans, lorries and motorbikes are at risk of their vehicles being blown over, it warned.
In Sheffield, the city council said it is aware of a number of trees being uprooted overnight and said arrangements are being made to remove them from the roads.
A Highways England spokesman said: "Road users in all regions are being advised to check the weather forecast and road conditions before they travel.
"Leave extra time if travel conditions are poor and delay the journey if the weather is too severe."
Network Rail is also warning of possible delays this morning.
A spokesman said: "Heavy rain and very strong winds have been forecast to affect parts of England, Wales and Scotland.
"Railway lines in areas affected by the worst weather may suffer disruption caused by falling trees and large branches, power cuts and debris being blown onto the track."
Met Office chief forecaster Frank Saunders said: "The low pressure system that is bringing these strong winds will move fairly swiftly from west to east over the UK and although there will be some disruption through Wednesday morning, the winds will ease by the afternoon leaving a day of blustery showers."
The Met Office said there is no connection between the high winds in the UK the severe weather battering the Caribbean and the US.