This sharpening was normally done within the regiment, by the armourer (often a Sergeant). However, as far as I can establish, there was no standard method of sharpening, and as a result (and of complete lack of sharpening) it seems that British soldiers often went into battle with unsharpened or poorly sharpened weapons. Adding to this problem, most British regulation swords were carried in steel scabbards, which helped to blunted even the best sharpened swords. This was mitigated to some degree by making the throats of some scabbards (where the edge rubbed the most) out of 'German silver' (like pewter), which was fairly soft, or by having wood or leather fitted inside the scabbard. In India some cavalry officers adopted leather-covered wooden scabbards and eventually these became standard issue.
The lack of attention to sharpening, coupled with scabbards which blunted egdes, led to the impression in some quaters that British swords were feeble in offence (compared to tulwars or other swords), and may also have helped lead to a preference of using the thrust (as even a blunt-edged sword would execute a thrust effectively). In fact, when British swords were kept sharp they often performed sterling service, removing heads and limbs, just like the feared tulwar. Surgeons who attented the field of Balaclava in 1854 reported hundreds of sword wounds.
Some officers ensured they had the best 'fighting swords' (often as a second sword - besides their regulation sword) by going to a quality manufacturer (like Wilkinson, Mole, Thurkle, Reeves, Pillin etc) and requesting special design details, like the patent solid tang, or a different blade section, and that their weapons be factory sharpened and supplied in scabbard that would not blunt them. British officers took many important lessons from Indian soldiers, including looking at how they sharpened their swords.
Below I will list examples of period text talking about the sharpness of swords, as I come across them.
He is a typical passage where well sharpened tulwars and poorly sharpened regulation swords are compared:
The Hindoostanee dress and equipment would doubtless have been more appropriate and serviceahle to these men during that war, than the uniform and arms of Heavy Dragoons. Chakeos, leather breeches, and jack boots might have been advantageously dispensed with, and a tulwar substituted for the heavy and blunt regulation sabre, whose rattling in the scabbard was to be heard at such a distance, as almost to prevent the probality of a secret movement. The troopers themselves were aware of the uselessness of these weapons, and consequently, made more use of the pistol than the cold steel. On one occasion, during a sharp skirmish, the Serjeant Major of the corps, with a willing heart, a strong arm, and the full sweep of his sword, nearly cut through the neck of a Burmah; the head and trunk retaining their alliance by merely a thin portion of skin. The Subadar-major pointed out this case, as an argument of the superiority of a tulwar. " Here," said he " is a fair cut, made with all the strength of a powerful European; the weakest trooper, with a Hindoostanee dress tulwar, would have severed the head from the body!"
From: The Meerut universal magazine, Volume 1, 1835.