The agency warned of the threat of vehicles becoming stranded, power cuts and villages being cut off.
The "be prepared" alert covers parts of the Central Belt and southern Scotland from 3pm until 8am tomorrow.
However, snow showers are now expected to continue across the whole country away from east coast until late on Friday.
A separate wind warning has also been issued, for gusts up to 80mph tomorrow night.
A Met Office spokesperson said: "Heavy, squally showers of snow will continue through Tuesday and Tuesday night, with longer spells of snow for a time, especially on Tuesday evening.
"Travel delays on roads are likely, as are public transport cancellations, and some vehicles may become stranded, more especially on higher routes.
"There is a chance of power cuts, while services such as mobile phone coverage may be affected.
"Some rural communities seem likely to become cut off.
"In addition roads and pavements will turn icy, increasing the chances of accidents or injuries.
"Over the hills, snow will drift in the strong to gale force winds.
"Some high ground is likely to see 15-25cm building up, especially above 200-300 metres.
"At low levels throughout the warning area, 5-10 cm will be more typical.
"Showers will be accompanied by hail and lightning at times, while strong winds bring the potential for temporary blizzard conditions and drifting of lying snow, mainly over high ground.
"A gradual improvement is likely on Wednesday for most areas."
The separate Met Office yellow - "be aware" - wind warning is for gusts of 60-70mph "quite widely" across the southern half of Scotland and "a small chance of winds reaching 80 mph in places".
It will be in force from 9pm tomorrow until 1pm on Thursday, and also covers the rest of the UK.
The spokesperson said: "Very strong winds will affect parts of the UK during Wednesday night and into Thursday.
"The strongest winds are expected to transfer eastwards across the warning area before clearing by early afternoon.
"Road, rail and ferry services may be affected, with longer journey times and cancellations.
"There is also a possibility that some bridges may close.
"Power cuts may occur, with the potential to affect other services, such as mobile phone coverage.
"Injuries and danger to life from flying debris are possible, along with some damage to buildings.
"In addition, large waves are expected to affect some western coasts with beach material being thrown onto sea fronts, coastal roads and properties.
"The location of the most damaging winds is currently uncertain, although parts of northern and eastern England along with southern Scotland currently look more at risk.
"Along with the strong winds a spell of heavy rain can also be expected, which could also be combined with a temporary thaw of lying snow in the north."