What a fascinating question! I think the answer is that they might have done but there is no record of them doing so. We should treat Ken Follett's statement as a work of fiction.

It's plausible but, to the best of my knowledge, not contemporaneously documented. In fact we know very little about the average medieval person's walking patterns in general and I'm not aware of any contemporary sources that detail this practice.

Follett says that this method was used when particular speed was required rather than as a norm, but he does indeed imply that it's something that everyone (in his world at least) is aware of.

Gwenda was more tired than she had ever been in her life.

She knew that the fastest way to cover a long distance was to run twenty paces then walk twenty paces. She had started to do that half a day ago, when she spotted Sim Chapman a mile behind her. For a while she lost sight of him, but when once again the road provided her with a long rearward view, she saw that he, too, was walking and running alternately. As mile succeeded mile and hour followed hour he gained on her. By mid-morning she had known that at this rate he would catch her before she reached Kingsbridge.

Footwear

There is an emerging "school of thought" that would present logical arguments against it being plausible, particularly claims by reenactors and the like (partic. Warzecha) that medieval inhabitants of lands around the Channel and North Sea would have walked on the balls of their toes rather than using their heels, something that they partly infer from what they say is the universally soft-soled nature of surviving medieval footwear.

However, we know very well that hard-soled footwear was in regular use in the Roman era and that by medieval times both wooden-soled footwear and "built-up" leather footwear were normally accessible. And we have good finds of perfectly suitable solid-soled medieval shoes.

Anthropological evidence finds little to support that idea of ball-toe walking either (Jacquet et al, 2016, Myska et al 2020) and instead supports the idea that a heel-favouring gait has been the norm for millennia. No specific evidence has been found amongst medieval populations that would support the idea of a hugely differing gait in European people during that particular period.

The Method Itself

As for the walk-run-walk practice itself, it is something that is known in the real world.

I learnt this technique as a young Cub Scout, it's straight from the writings of Lord Baden-Powell, he of the retrospectively badly-titled Scouting For Boys. The method is considered to be Baden-Powell's own invention, although it illustrates the times to say that it's acknowledged he copied the method from Zulu runners that he'd observed during his time in Southern Africa.

A successful demonstration of the pace involved covering a mile in 12 minutes, which would represent an average speed of 5 mph. I did mine in the rain but, being British, that was largely inevitable.

The method's use by Zulus, experts at navigating large distances through some pretty hostile terrain suggests that it does indeed represent a good balance of speed versus physical efficiency. It should be said that I know nothing about Zulu culture and that I'm taking Baden-Powell's account of his observations to be accurate. If you were interested in knowing how long Zulus had been travelling that way then that would be an interesting research aside :)

Summary

Did medieval people usually travel this way, or even know how to? The footwear existed, the gait existed, and humans somewhere in the world had possibly already figured the practice out.

There are no contradictions from historians of Baden-Powell's claim to have "invented" or introduced the practice to the West which lends likelihood to there being no documentary evidence of earlier common practice in Medieval times or otherwise.

So the best answer is... they might have done but there is no record of them having done so.