I'm not going to claim to know a lot about English military history or tactics, but there are a few things I've noticed. They were reliant and very proud of their heavy cavalry (and massive numbers of rapid-firing longbows of course) for head-on charges instead of flanking runs. I think they used this as a dual purpose thing, both for direct damage and for the demoralization effect that it had when they trample the front line with horses (causing the weaker-willed soldiers to flee in terror). There's lots of evidence both in writing and artistic depictions (particularly tapestries and stuff) that show their horsemen trampling the enemy with the infantry and archers behind them, which to me suggests that they were used a lot for head-on attacks. The archers were more likely to go on flanking runs, as there is existing writing of longbow military doctrine for acquiring an enfilading fire angle to inflict maximum casualties on the enemy line even with the inaccurate longbow.
If you remember the movie Braveheart, they were so arrogant that their heavy cavalry could simply overrun William Wallace's light infantry units. I know there's some historical accuracy to the use of makeshift pikes to counter the heavy cavalry, or at least as internet resources claim. Apparently the pike was one of the keys to the Scottish resistance being successful against the seemingly invincible heavy cavalry of the English at the time (in the movie I remember them saying that no army has ever won against their heavy cavalry without having cavalry of their own, probably for couched lance confrontations in the open field), as it was cheap to make and easy to train in (easier than a sword anyway), which was perfect for the peasant armies.
I would have to say though, based on the other opponents that the medieval English armies were used to facing, they didn't deal with pikes a lot, and didn't really fear pikemen so much. I remember reading that the French used pikes a bit, but not as a primary infantry doctrine. The English themselves didn't really use pikes as an infantry weapon or have a pikeman division until much later than the medieval era, which Warband is loosely based on. Correct me if I'm wrong on any of this.
I'll have to research a bit more because your question got me curious. But for the most part, there's no reason you shouldn't use your heavy cavalry for flanking or frontline charging. In Floris we have quite a few pike-heavy infantry formations. Even the Sarranids are a little scary if you're fighting a bigger army. I looked at their front line and was like, "Oh it's just Al Haqanni. They're only tier 4 units and their shields are the size of dinner plates." I charged my horses in then blam, stopped by desert pikes. I got knocked out in that battle, and I learned from then on to get rid of the pikemen first before charging the horses in.
As you said the terrain will play an important role in your decision-making for this. I only send infantry-only or light cavalry flankers if I know they'll have some kind of cover like a hill or something, or if I can keep their archers firing on my front line instead of them. For all the variables of combat, this is why I have to stay out of the battle myself and stay on a horse and even ride up to a high point of the battlefield, just to make sure I can direct the tactics better.
But as a matter of personal preference, there's a certain manly feel to it when you line your heavy cavalry and infantry up as a frontline charge, especially if you're right there with them lol. That's why for Swadia I typically play in this doctrine, much like how I play with fast light infantry and cavalry with Sarranids: not the smartest way to build an army, but it sure feels good when it works. I sometimes get an adrenaline rush when their archers are taking a few of us down or arrows zip by my head, but I slowly push my line forward until the opponent has no choice but to retreat. Because once that front line gets in there, the battle is definitely over, even if they call in reinforcements. Did you notice that reinforcements are generally lower tier troops that look like sad, sad peasants? By the time they've called in some reinforcements, my front line is already upon them and it's just so sad to see how many of the enemy is dying just like that lol. I feel bad sometimes and want to grant them safe passage if they drop their arms and go back to farming or something. But they keep fighting, so we have no choice but to slaughter them. Even after they flee they sometimes reform their ranks and try to throw stones at us...