The phrase the Beast from the East, as used in British English, is defined as follows by the Met Office, i.e., the national meteorological service for the United Kingdom:

The ‘Beast from the East’ is a phrase used to describe cold and wintry conditions in the UK as a result of easterly winds from the near continent.
When pressure is high over Scandinavia, the UK tends to experience a polar continental air mass.
When this happens in winter, cold air is drawn in from the Eurasian landmass, bringing the cold and wintry conditions that give rise to the ‘Beast from the East’ moniker.

—Cf. also Siberian Express.

Although probably based on the phonetic resemblance between the nouns beast and east, this phrase may also have been influenced by the Beast from the East, one of the nicknames given to the former Russian boxer Nikolai Sergeyevich Valuev (born 1973), who competed from 1993 to 2009.

The earliest occurrence of the phrase the Beast from the East that I have found is from the following article by Nathan Rao, published in the Daily Express (London, England) of Saturday 12^th February 2011:

‘Beast from the East’ roars in with more snow

Winter is set to roar back to Britain with a vengeance this week as a Russian weather front threatens another big freeze.
And there will be no let-up until well into March, say weather experts.
In coming days, the “beast from the East” could bring snow chaos similar to the pre-Christmas whiteout.

In American English, the phrase the Beast from the East is occasionally used of similar cold and wintry conditions. The earliest occurrence that I have found is from The Times and Democrat (Orangeburg, South Carolina) of Saturday 24^th November 1984:

‘The Beast From The East’ causes flooding, erosion
Charleston, S.C.—“The Beast From The East,” northeasterly winds that send abnormally high tides landward, flooded city streets Friday and caused erosion, sea wall damages and flooded roads in nearby Folly Beach, authorities said.

On the pattern of the Beast from the East, and based on the phonetic resemblance between the nouns pest and west, the phrase the Pest from the West designates, in British English, a blast of mild air from the Atlantic, which brings strong winds and heavy rainfalls.

The earliest occurrence of the Pest from the West that I have found is from the following article by Nathan Rao, published in the Daily Express (London, England) of Wednesday 12^th December 2012—in which the Beast from the East also occurs:

Get your skates off! Now for lashing rain and gales

Britain’s big freeze will give way to an onslaught of wind and rain as the chaotic winter weather rages on.
The dramatic turnaround will be triggered by a blast of mild air from the Atlantic which will bring a week of snow, torrential rain and fierce 85mph winds, forecasters warn.
The South and South-west regions hit by devastating floods last month are once again in the firing line with more than an inch of rain expected in a few hours.
It comes as the UK is braced for the coldest temperatures of the year so far today when the “Beast from the East” bares its teeth.
Temperatures could plunge to -15C (5F) before an Atlantic low pressure system rolls in tomorrow.
Jonathan Powell, forecaster for Vantage Weather Services, said: “Wednesday is going to be pivotal in terms of the cold weather with biting winds making it feel even colder.
“Temperatures could well hit -15C in rural areas, with widespread frosts and ice causing travel problems.
“Then we have an Atlantic system bringing torrential downpours with the threat of flooding again. There will be fierce winds around coastal regions which could reach 85mph.”
There were warnings yesterday that more than 200 people could die every day in the cold.
The Department of Health said the number of cold-related deaths increases ‘substantially’ during severe weather.
The Met Office has a Level 3 cold weather health alert in place across virtually all of the UK until tomorrow afternoon.
It warns of a 90 per cent probability of severe cold, ice and heavy snow which could increase health risks to the elderly and vulnerable. Last night it issued severe weather warnings for rain in the North and the South-west on Friday.
The Met Office said “the Pest from the West” could bring more than an inch over high ground. Forecaster Dan Williams said: “When the milder air hits the cold air, there could be some snow on Friday which could cause travel disruption.”