The article below doesn't mention the role played in Woolwich but it does talk about the formation of the RE. It is taken from Wonders of World Engineering Part 40. 1937.
(and - by the way - the Royal Engineers have an excellent museum in Gillingham)
ENGINES of war are no new invention, and attempts at the mechanization of armed forces were made many ages ago in man's struggle for power. The scythes on ancient British chariots, the siege engines of the Romans, the armour of medieval knights, the application of gunpowder- all represent steps in the slow development of warfare through the centuries.
Now the day of the sword and spear has ceased and the defence of countries depends on the accuracy of scientific calculations and on industrial resources. The struggle now lies on either side between men and machines, and military engineering is of as much importance as-the courage and resolution of the fighting units and the efficiency of industry mobilized for war. In the British Army the vast amount of engineering work entailed in the theatre of war, from trench construction to road and railway building, is undertaken by-the Corps of Royal Engineers.
Among the earliest examples of military engineering in Great Britain were the great roads built by the Romans to facilitate the movements of their legions. To-day, road making is one of the most important of all military engineering activities in time of war. Mechanization can have little value without good roads.
Earthworks and other forms of fortification called for specialized attention even in the earliest times. By the beginning of the eighteenth century, after the establishment of an English standing army, a number of "King’s Engineers" had been appointed for special duties, mainly in connexion with fortifications. In 1700 there were twelve of these officials (they did not hold army rank) in the British Isles, but long before then the country's wars had necessitated the services of “Train Engineers" for the operation of the cannon and the supply of ordnance.
The artillery, however, was separately established in 1716 and a Corps of Engineers was formed with permanent personnel, augmented in war time, of twenty-eight engineers. A Company of Soldier Artificers was formed at Gibraltar in 1772, and these men played an important part in the siege of the Rock (1779-83) and in the building of the first of the famous gun galleries.
The Corps of Engineers was granted the title of "Royal" in 1787 and in that year also was formed a Corps of Royal Military Artificers, in which the Soldier Artificers were afterwards incorporated. The Artificers were responsible for Wellington's famous fortifications, the Lines of Torres Vedas, before Lisbon in the Peninsular War. In 1813 the Royal Military Artificers were renamed the Royal Sappers and Miners.
A private soldier in the Royal Engineers is still termed a sapper and a sap is the name given to a heading run out from a trench dug parallel with a line of fortifications to be attacked. The Royal Sappers and Miners performed invaluable work at the siege of Sebastopol and elsewhere during the Crimean War. At the conclusion of hostilities in 1856 the Royal Sappers and Miners (which consisted of rank and file only to this date) were united to the existing Corps of Royal Engineers. The East India Company's Engineer officers, with the traditions of Lucknow and the storming of Delhi well established in British history, were amalgamated with the Corps of Royal Engineers in 1862.