Synchronized movement was a key aspect to military units like the Roman legion, the phalanx, and later musketeers.

The legion worked because the Roman armies were disciplined war machines that were constantly training in order to be able to execute such precise movements.

In the 16th century, the Swiss pike phalanx devastated armies heavily invested in heavy cavalry. One of the reasons these infantry were able to defeat cavalry was because they were heavily disciplined to stay together as a group. This meant moving tightly together to prevent cavalry from being able to sweep in and disrupt the formation.

The musketeers and other infantry that utilized gunpowder before and during the Napoleonic Wars had limited range. For a musket to reliably hit a target, it had to be fired rather close to the target. Another way to guarantee the target was hit was to fire a lot of muskets at the target. When a lot of men are put together with gunpowder weapons in a tight space, discipline is key to allow the military unit to be effective, so the men would be highly disciplined. When moving, they would move as a unit.

This question can only be answered for a specific time and type of unit. European armies after around 1700 fought in something called linear tactic. In this time, regular army units were indeed drilled to march in a uniform speed (cadence) and follow commands strictly and promptly. Commands could be given verbally, but also by drums, trumpets or other means.

It goes without saying that in battle, order can not always be maintained. Infantry can charge, the ground can be uneven or broken, units can be broken by artillery etc. Maintaining unit cohesion and a uniform pattern nevertheless was an ideal during the 18th century, and great effort and time was invested into achieving this goal.

Other military formations that relied on strict unit cohesion were the swiss Gewalthaufen or the spanish Tercios. In these cases, keeping a wall of polearms pointed to the enemies was vital to defend against cavalry charges. These units were not as drilled as later line infantry, and would not be able to change directions (pivot) as easily.

Compared to that, an army from the early or high middle ages would not have been as orderly. However, sticking close to your comrades is a very natural thing to do in battle. Marching at your own speed will get you isolated, and getting isolated was always deadly. Unless an army had been routed, it would therefore always maintain a certain amount of cohesion - but not nearly as much as the well-drilled (professional) armies of later times.

I'm not much of an expert on the American revolution but during the Middle Ages this was definitely not the case. During this time armies on campaign were organised around the 'great men' who levied them and paid their wages. These lords would normally have men-at-arms in proportion to their wealth and rank and while these warriors were highly trained and maintained high levels of small-unit cohesion they were heavy cavalry and as such didn't do much in the way of marching. The infantry forces were something of an afterthought and, being raised as individuals or in small groups at the start of a campaign, had nowhere near the level of uniform discipline and organisation to march in time. The Swiss Pikemen in the 14th century were the first force of infantry in Western Europe to march in time since the Roman legions 1,000 years before.

Towards the end of the Middle Ages and into the Modern Period the increased importance of gunpowder weapons on the battle field and the rise of more efficient, centralized government (which facilitated the creation of standing armies) saw increased importance placed on discipline and organisation. This culminated in the 18th century with the armies of the Ancien Regime which placed an obsessive amount of emphasis on drill, standardization and discipline. Kind of a side note: but iirc correctly European forces in the colonies actually operated very differently to how they did on the continent. Most of what I know of this period is from the Napoleonic Wars in mainland Europe so I can't be sure but I think that European forces in the colonies made extensive use of local militias and light infantry due to the comparative infrequency of pitched battle as well as the logistical difficulties of transporting a European army across the world.

At least part of Frederick the Great's tactical success was because he was the first to drill his regiments with a cadenced march. Until the Seven Years' War, nearly every other European army marched their troops in route step. Consequently Frederick's army was much faster in changing formation than the Austrians or French, who might take as much as two hours to move from column to line.

Source: Paul K. Davis, Masters of the Battlefield