Could it be Rick Raphael's "Code Three"? That has police on a megahighway and it did run in Analog.
Here are some excerpts from a review on Goodreads (bold emphasis mine):
The North American Continental Thruway is a road system that stretches coast to coast across the USA, and from the southern-most point of Mexico all the way up to Alaska. Each 'artery' is five miles wide. Yes, five miles wide! With different lanes for the different rates of speed the almost-flying cars of the time can maintain. Naturally, there has to be some some of control and patrol on such a thruway, and that is where Beulah comes in.
I love Beulah. Listen to this: She was sixty feet long, twelve feet wide and twelve feet high; topped by a four-foot-high bubble canopy over her cab. All the way across her nose was a three-foot-wide luminescent strip. This was the variable beam headlight that could cut a day-bright swath of light through night, fog, rain or snow and could be varied in intensity, width and elevation.Beulah had weapons to meet every conceivable skirmish in the deadly battle to keep Continental Thruways
fast-moving and safe.
So, as you can see, it is a police cruiser of gigantic proportions. It does travel through the USA and Canada (the Baen short story anthology I linked above has them starting by Alberta). The ISFDB entry shows that it was published in Analog in February of 1963 in the US and June of 1963 in the UK. I've found found a few references in reviews online stating that it appeared in 3 segments, so that matches.
It's been released as a short story and as a novella. It looks like there was a sequel, "Once a Cop", where the emergency technique developed at the end of the first book is made standard, increasing the danger for the police even more, leading to some degree of meditation on the citizenry's constant need for more speed despite how dangerous it makes their lives, and just how far the police are willing to go to keep people safe.