What is the usual flight path for airliners flying from London GB to Vancouver, Canada; or Seattle USA?

Is it due west across the Atlantic ocean and the American Continent, or due north and across the Arctic regions.
A flat map would suggest a transatlantic route but as the real earth is a sphere a transpolar route could well be quicker.
Also routes to other airports in the west of Canada and America.

Looking at a 2-dimensional map, the flight path is an arc, taking you over Greenland

Generally over the southern part of Greenland. Scroll to the bottom of the article below for London to Seattle route.

A flat map is a two dimensional representation of a three dimensional object. A Great Circle is the shortest distance between two points on the surface of a sphere. Long-distance flights try to follow the route as close to the Great Circle as possible. Calculating the departure and arrival headings for a Great Circle route requires special kind of math called Spherical Geometry.

While the actual routing may vary day to day or even hour to hour, a typical flight will head north over Scotland, then Iceland, across central Greenland and then over Nunavut, Northwest Territories, northern Alberta and arriving at its destination from the north-east.

Most transatlantic flights will take the route that is shortest and reduces the need to fly for long distance over open water, hence they mostly opt to fly north to Greenland across the upper USA lower Canada and follow the coast down to Seattle and beyond o San Francisco and Los Angeles

A flat map would suggest… True it does.

as the real earth is a sphere a transpolar route could… Be shorter, could be over an area not covered by Air Traffic Control, have alternate runways to use in case of emergency, have rescue services nearby…
International flights have been using Polar routes since World War TWO.
You can easily get a flat polar projection map to see this.
On line google map will show the route.

Just south of Iceland, over Greenland, Canada ( Nunavut Territory, Saskatchewan, Alberta, British Columbia)
Washington State for Seattle.

For a California destination the route over Canada would be over Manitoba

Google maps measure distance function a right click shows you all the routes you care to look for.

ATC decides the exact route taking into account other Air traffic, weather and other stuff.
In the old days when planes could not fly as far. The route was strait line from London to Gander Newfoundland for a refuelling stop, To reach the west coast of North America would usually require one more fuel stop along the way.