The first named storm of the year hit the country overnight and will continue through the morning, 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 the vehicles being blown over.
The M48 Severn Crossing in South Gloucestershire is closed in both directions between junctions 1 and 2 due to the gales, while the Humber Bridge in East Yorkshire has been shut to high-sided vehicles.
The Ouse Bridge on the M62 between junctions 36 and 37 has also been closed.
A 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.”
Motorists are also facing road closures up and down the country due to fallen trees, the Press Association reported.
Police forces in Staffordshire, Cheshire and Gloucestershire have all reported trees being blown over by the winds overnight.
There is also likely to be disruption on the trains during the morning rush hour.
Speed restrictions have been put in place on some lines, including those travelling between Eastleigh and Weymouth and between Petersfield and Portsmouth, as well as between Welwyn Garden City and Arlesey.
A spokesman for Network Rail 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.”
Meanwhile, power companies are reporting power cuts overnight, affecting huge swathes of the country.
More than 800 homes were without power in Nottinghamshire, while 700 homes were affected in Lincolnshire.
Parts of Wales and the South West also reported power cuts.
Dozens of people took to social media overnight to complain that the strong winds were leading to sleepless nights.
The Met Office amber weather warning for wind is in place until 6am on Wednesday.
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 was no connection between the high winds the UK is expected to see and the severe weather battering the Caribbean and the US, with the UK’s weather system originating north in the Atlantic, independent of the current hurricanes across the ocean.
As Storm Aileen clears out eastwards into the North Sea, the UK will be left with cool, showery conditions by the end of the week and into the weekend, the forecasters said.
Rod Dennis, spokesman for motoring organisation the RAC, said: “Aileen’s impact is likely to be short but sharp in the north of England, and felt most by those driving into the morning rush hour.
“High-sided vehicles will need to take particular care, but anyone driving can expect to experience some very strong gusts and heavy rain which will make driving conditions difficult.
“Plan ahead and if you can delay your journey until conditions are expected to improve later on Wednesday, then do so.”