Okay, there are three different avenues to this question: (1) how do people walk today, (2) how much is our gait cycle pre-determined and constrained by the skeletal anatomy of our species, and (3) did people in 1500 have a different walking pattern for whatever reason. I'm moderately qualified to answer the first two parts, and will leave the third to a proper historian of the period who can discuss written and artistic representations of walking. Here we go...

First, physical therapists are the masters of modern gait analysis. They spend a significant portion of their academic training observing gait patterns, and several state of the art motion sensing labs dedicated to analyzing gait routinely publish in the field.

Here is a good video explaining how experts divide the gait cycle. Gait is separated into two phases, stance phase (when your weight is on the planted foot, roughly 60% of the gait cycle), and swing phase (when the foot is in the air, swinging forward for another stance phase, roughly 40% of the gait cycle). Stance phase begins with first contact/heel strike then progresses to loading response then midstance and finally toe off/terminal stance as the foot moves into swing phase. Check out this image for an idea of what muscles fire at what point of the gait cycle. Modern physical therapists generally take heel strike when walking as the norm.

Of course no two people are completely alike. Pathologies/injuries can modify the gait cycle, and various modern subcultures occasionally adopt culturally atypical gait patterns for a variety of different reasons, including occupation (such as the scissor gait pattern of runway models). The question then becomes, cross-culturally do we walk in a similar pattern today? Specific to the video you posted, do humans with less sedentary lifestyles, or those who spend most of their time barefoot, make first contact with their forefoot or with the heel?

One of my favorite examples of well visualized running and walking gait during extreme exertion comes from the Life of Mammals BBC documentary which managed to capture the San of the Kalahari on a persistence hunt. The hunters were encouraged by the documentary field crew to showcase a style of hunting that involves chasing prey for extended periods of time until the prey overheats. This specific hunt took nearly eight hours from start to finish. As you watch the video you can see the San hunters heel strike when walking, but transition to forefoot strike when running (more on this later). Even when stalking the prey the showcased hunter prefers a slow motion heel strike gait pattern rather than a forefoot first contact. Granted he is in shoes so we need to examine the gait pattern from a culture where walking is predominantly barefoot.

Territorial encroachment, violence, and resource destruction are stressing previously isolated nations along the Madre de Dios River in Peru. In the past decade the foraging populations who were previously isolated from almost all contact with sedentary populations are now starting to show themselves along the river. A recent video of one of these initial contact events show Mascho-Piro/Nomole walking toward the camera. As the video shows, they make first contact with their heel, not with their forefoot. This is a culture that, as far as we know, shows no evidence of footwear, and who engage in barefoot walking for their entire lives. They heel strike when walking at a relaxed pace.

So, given that modern humans seem to generally walk the same way do we know when this gait pattern emerged? Put another way, when did hominins become bipedal, and is there evidence of a specific gait pattern in the paleoanthropological record?

Bipedalism emerged roughly 6-7 million years ago (mya) with either Sahelanthropus tchadensis or Orrorin tugenensis. The transition from knuckle-walking ancestor to a bipedal form required anatomical changes. Modern chimpanzees use ~75% more energy when attempting to walk bipedially, and that extreme energetic cost meant something had to give. By the time of Ardipithecus ramidus 4.4 mya bipedal gait appears to be used extensively and we start seeing major skeletal changes to accommodate the new locomotive pattern. As hominins became obligate bipeds the foot, knee, pelvis, thorax, spine and even the skull needed to accommodate to new biomechanical demands. The calcaneus (heel bone) increases in size and shape, the grasping toe is lost as the great toe moves in line with the other digits. An arch develops in the foot, the pelvis becomes more bowl-shaped instead of elongated, and the foramen magnum (hole for the spinal cord) moves under the skull instead of its previous posterior position. All these changes are well in place by 3.6 mya when the famous Laetoli footprints were left as an Australopithecine walked across volcanic ash. Bipedalism, and the skeletal anatomy to make it energetically feasible, continued to be refined with the emergence of the genus Homo. Based on the best available evidence from fossils and preserved footprints, a heel strike pattern appears to be the norm for our genus for several million years. The recent discovery of the 800,000 year old Happisburgh footprints records the meanderings of a group of Homo antecessor and reinforces a thoroughly modern gait pattern with a strong heel strike seen in a 3D reconstruction.

As the gentleman in the video states, "it takes less energy to be a heel walker than a ball walker" and it looks like we inherit a suite of skeletal changes that have predisposed us to a heel strike gait pattern for at least several hundred thousand years.

Now, why might he be mistaken in this strange forefoot/heel strike debate? First, barefoot humans walk differently on different surfaces. If the floor is littered with stray lego blocks, or you are walking on a rocky shore, a guarded forefoot pattern can protect from damage. However, walking barefoot habitually causes deposition of extra skin to protect against that damage. Anecdotally, when in the jungles of isolated Peru my indigenous friends could happily walk barefoot over surfaces that made my pampered, shoe-clad feet, recoil. There will be room for debate, but I would challenge the assumption that Medieval walkers spent more time walking on harsh surfaces than moderately smooth surfaces that their feet accommodated to. Second, there is a big difference between running gait and walking gait. In the video the gentleman is discussing walking gait while showing forefoot strike of runners. Bramble and Lieberman made waves when they proposed endurance running was the prime mover for the emergence of our genus, and gave rise to the "born to run" movement. The authors, and subsequent research, suggest forefoot striking while running conserves energy and places less biomechanical strain on the body than heel striking. I'm not sure if this gentleman heard of this running debate, and modified it for Medieval use by assuming a forefoot first contact when walking.

All this said, I am not a scholar of Medieval Europe. Cultures are strange and people can modify even the most intrinsic human actions based on societal norms. The best guess, based on modern movement patterns and our evolutionary history, would suggest a normative heel strike gait pattern for most walking conditions.

To add to anthropology_nerd's very thorough answer, you know how ancient Egyptians were always drawn in a very distorted posture that's painful to even try to twist yourself into? Just because something's a theme in art doesn't mean it was a reflection of real-life. One of the things art historians have to learn (if they're the sort who try to make real-life postulations from art, at least) is how to distinguish what is and is not artistic license. I think the guy in the video's reading a bit much into an artistic style without considering that body mechanics, conservation of energy, and the like all point very strongly towards heel-walking (but ball-running).

Additionally, because period illuminators were not biologists, they often drew things wrong from a biological perspective without even intending to make a stylistic choice simply because they weren't thinking about why it wouldn't be right. Look at all the creepy Renaissance Jesus babies that look like tiny adults. Or the evolution of elephants in Western Art, drawn by people who'd clearly never seen one in real life. No one's suggesting Renaissance babies came out of the womb weirdly muscular and with tiny heads, or that elephants used to look like tapirs, litopterns, and trumpet-nosed Dr. Seuss animals, because that'd just be silly. You're question isn't obviously wrong like those are, but the whole illuminator mistake/style thing needs to be ruled out before seriously considering it to reflect a real bizarre and drastic ephemeral shift in a human population's physicality.

Side question. Did boots during this time resemble the boots today? I hear they were more like thick leather socks.