Northern Ireland and southern Scotland will be worst hit by the storm, with warnings from forecasters of the potential for "injuries and danger to life from flying debris".
A yellow warning for wind has been put in place from 12am until 3pm on Sunday, in the two regions.
The Met Office said: "There is a small chance of damage to buildings, such as tiles blown from roofs, with the potential for injuries and danger to life from flying debris."
Forecasters added that large waves and beach material could be thrown onto coastal areas, which could present a hazard. Power cuts and mobile phone signal coverage problems could also occur.
"There is a small chance of longer journey times or cancellations as road, rail, air and ferry services are affected, with the chance that some roads and bridges could close," according to the Met Office.
Sky's Joanna Robinson said: "Friday brought snow and ice issues to parts of the north, but it's wind and rain that may bring some disruption over the weekend.
"Storm Dylan is due to move in on Saturday night, bringing potentially damaging gusts to parts of Ireland, Northern Ireland and southern Scotland.
"The strongest winds will be on the southern flank of the low pressure system, with gusts up to 80mph.
"Heavy rain will be an additional concern, with local flooding likely, especially where the ground is already very wet across south-west Britain.
"The unsettled theme will continue into the new year."
In south-east Wales, south western, central and southern parts of England, heavy downpours are expected across the weekend.
There is a yellow warning for rain in place from 6pm on Saturday until 9am on Sunday, with 25mm predicted to fall, rising to 40mm in some places.
The Met Office said: "With the ground very wet in these areas, this is likely to lead to rising water levels and some flooding in places."
The end of year battering comes after snow fell across much of the UK on Thursday night and into Friday, causing road and rail disruption as well as problems at some airports.
Glasgow had the biggest snowfall, with more than 10cm (4ins) in some parts of the city.