The Met Office has issued two yellow warnings for high winds covering Northern Ireland, Wales, most of England and the west coast of Scotland.
Dozens of flood alerts have been issued by the Environment Agency in England, including two warnings for the River Ouse at York.
Gusts of up to 80mph are being felt in western Scotland, Northern Ireland and the northwest of the Republic of Ireland.
Winds of over 60mph swept widely across western parts of the UK in the early hours of Wednesday, after gusts of up to 75mph were felt in Scotland on Tuesday night.
Commuters using trains in Wales, Scotland, the north of England and East Anglia were warned that services could be disrupted by high winds during the morning rush hour, with speed restrictions in place across a wide area.
Trains between Durham and Newcastle were halted until at least 9am after overhead electric wires were damaged, impacting LNER, CrossCountry, Northern and Transpennine Express services.
Some Virgin Trains services between Manchester Piccadilly and London Euston and some trains between Glasgow Central and Preston were cancelled.
There were also reports of trees blocking roads and some exposed routes in the northeast of England being closed to high-sided vehicles.
Heavy showers with some thunder are expected across the northern half of the UK as the storm moves across throughout the day.
Other transport links are at risk of being impacted as Storm Gareth is expected to continue moving across the country throughout the day.
The worst of the winds were predicted to occur in the early hours of the morning, before gradually easing in the afternoon as the storm moves out to the North Sea.
Racing at the Cheltenham Festival could also be affected.
Punters have been advised that organisers would be holding a precautionary inspection in the morning to assess conditions for racing.
Three climbers died on Ben Nevis as an avalanche tumbled down the mountain after Storm Gareth blew in with strong winds on Tuesday.
Six crew members were airlifted from a French fishing boat off Land's End, Cornwall, in "atrocious" 20ft waves and storm-force winds later that night, the Maritime and Coastguard Agency (MCA) has said.
HM Coastguard was alerted at around 10pm that a 79ft fishing vessel, named La Fanette, had suffered engine failure.
Another fishing vessel went to help and a lifeboat was launched.
Met Office forecaster Alex Burhill said: "We've got some very strong winds.
"The strongest in the morning rush hour are probably going to be around Northern Ireland and also western coastal parts, particularly western Scotland and north-west England.
"Eastern areas might start getting some very strong winds by that point, but generally across much of the UK it's going to be very windy this morning.
"People need to be aware that there could be some travel disruption because there could be delays to things like ferries and aircraft and also bridges may need to be closed."
Storm Gareth swept in across Northern Ireland on Tuesday, causing problems on the roads with fallen trees and the cancellation of ferries.
Met Eireann, the Irish Meteorological Service, also warned of "severe and damaging gusts" as the storm moved over the country.
The storm is expected to have moved out into the North Sea by Wednesday evening, although another band of wet and windy weather is expected to arrive from the west on Wednesday night.