Finding The Dominant Problem
Don't underestimate the zombies.
In most situations, the key to predicting the consequences is to identify the one or two dominant impacts that a situation will have on a population and to ignore all third and higher order effects.
Lots of the problems that other answers to this question have solved are basically third and higher order effects, however.
The single greatest threat posed by a medieval zombie apocolypse is the threat that they would pose to farmers doing their work and to crops in the fields, as explained at greater length below, something that other answers haven't given much thought to addressing.
Less Critical, Later Order Problems
Keeping zombies at bay in fortified walled cities, castles, monasteries, towns and even enclosed country manors, and fortified caves might not be much of a problem.
Solutions like moats would be popular (and for backstory, Copper Age and Bronze Age Iberia was full of walled enclosures that often, perhaps even usually featured moats, so perhaps zombies were a particular problem in Portugal and Spain for some reason).
England has abundant supplies of water that would make it unnecessary to be vulnerable around fresh water supplies.
And, defeating zombies militarily in planned engagements with armored soldiers and calavry might work, against an ill armed, not terribly fast moving, ill coordinated foe, however deadly they might be if their infection is passed on.
But, feeding everyone during the zombie apocylopse would be a much more daunting task.
Feeding The People
Most People In The Middle Ages Were Poor Rural Farmers
In the Middle Ages, the vast majority of the population (70%-90%ish) was engaged in farming as serfs on feudal estates and herding on unenclosed pastures. And, it is not practical to have a very large share of workers in a field wearing armor or carrying weapons in addition to farming tools while working the fields. Probably only something like 1% to 5% of the population could be spared as anti-zombie soldiers.
Also, only a small minority of people could afford to use horses for transportation, so most serfs and free peasants would have to live pretty close to the fields that they work, greatly limiting the number of people who can live in fortified urban centers. Recall that feudalism was necessary in the first place because kings lacked the logistic capacity to rule their kingdoms directly, rather than through intermediary noblemen who lived cheek by jowl with their subjects. High population densities were simply not economically or technologically viable for the most part.
In short, there are real limits to the viability of fortress/quarantine solution.
Given that zombies aren't particularly fast, probably the best strategy for farm workers would be to put some older children who are still relatively small and not too useful in hard labor in tree tops or artificial lookouts to keep their eyes on the lookout for zombies, and have them signal if they saw any coming, so that everyone could jog to the nearest farm house or purpose built rural zombie shelter before the zombies arrived and from there ring a loud bell or drum to summon assistance from the local lord's soldiers. The strongest farm workers would form a rear guard with their tools as improvised weapons, as the children scrambled from their lookouts and fled for safety at a higher speed than some of the older farm workers, to join the less fit farm workers fleeing while the rear guard covers them.
Ley lines historically were probably established deliberately by prehistoric Britons as a series of watchtowers connected by line of sight to provide an early warning system and this too could be integrated into the zombie alternative history.
How Can Crops Be Kept Safe?
There is also the problem that crops in the fields are vulnerable to trampling by zombie hordes, even if the zombies aren't doing so intentionally and have no use for the crops themselves. Farming requires sustained peace and order for many months at a time, because the growing crops can't be moved until harvest time two or three times a year, leaving months at a time when they are vulnerable not just to weather and animals but to destruction as wandering zombie hordes trample them.
It is all good and well to hide in a castle or rural zombie shelter or fortified farm house or manor for a few hours or couple of days while the zombies in the neighborhood move along. But, if you hide in your fortified place of safety while your crops are ruined, the zombies will have killed you as sure as if they had actually eaten your brains. Only starving to death takes longer and involves more sustained suffering as you watch children and older people die first, and those in between struggle to hold on with food shortages that make them vulnerable to diseases and zombie attacks.
Yet, it would be far too expensive to grow any crops but a few high value ones (such as herbs and spices and maybe some berries or orchards or vineyards) or that don't need light and can be grown in cellars (such as mushrooms) within fortifications.
If the English people are going to survive, they need better solutions than high strong walls that are expensive to contain large areas with in order to protect their crops.
Swamp Buffers For Fields
One attractive solution to protect more everyday crops would be to drain the interior of bogs and moors and swamps, while using undrained perimeter areas as wet buffer zones to protect the crops from wandering zombies. Alternately, rivers and streams could be intentionally diverted to flood perimeter zones around farmers' fields, a bit like Asian farmers flood rice patties, to create new buffers and wetlands (with the downside of providing a home to insects and other vermin who also pose a public health threat). Maybe access could even be via a short flat boat trip or a plank walkway that could be pulled up if zombies were sighted.
Big Cats To Protect Fields
Another way to keep fields safe would be to encourage top predator species (felines would be preferred because they don't eat grain themselves) to patrol your territory, at least if they could prey on and/or eat zombies. Lions and tigers could come to be known as protectors of the land, rather than merely as threats to livestock, and might learn to prefer to hassle zombies who would generally be unarmed, not as fast as them and relatively stupid, to trying to make uninfected humans their prey. This could also provide evolutionary insights into why zombies move faster at night when large felines are more active, than during the day.
If you could find a sustainable way to grow food free of significant zombie interference, life could go on, but I'd suspect that you'd see rapid population collapse primarily due to impaired horticultural production, rather than direct losses to zombie infection and attacks, in places where a quick solution to keeping zombies out of the fields could not be devised fast enough.
This population bust would impair high culture and would strengthen the hand of serfs vis-a-vis their feudal masters, because there wouldn't be enough serfs to tend all of the land. Serfs would gain more freedom and rights during the zombie era in England. A zombie outbreak might cause democracy to emerge earlier than it would have otherwise.