Eton, scholar of King's College, Cambridge, 1740, where he took his B.A. degree in 1744, and subsequently a fellow of his college, and M.A. 1749. Meanwhile, instead of taking holy orders as his friends had intended, he obtained an ensigncy in a regiment of foot then commanded by Lord Henry Beauclerk (afterwards 48th foot, now 1st Northampton), on 26 March 1744 (Home Off. Mil. Entry Book, xvii. 466). Beauclerk's regiment, of which Henry Seymour Conway [q. v.] was afterwards colonel, was present at Culloden 16 April 1746, and on 21 May following Draper was appointed adjutant of one of the battalions of the Duke of Cumberland's own regiment, 1st foot guards, in which at first he held no other rank (ib. xx. 249). He went to Flanders with the 2nd battalion 1st guards in January 1747 (Hamilton, Hist. Gren. Guards, ii. 141), and became lieutenant and captain in the regiment 29 April 1749 (ib. app. vol. iii.). He appears at one time to have been aide-de-camp to the second Duke of Marlborough when master-general of the ordnance (Gent. Mag. xxvi. 44), and on 23 Feb. 1756 married his first wife, Caroline, second daughter of Lord William Beauclerk, brother of his old colonel and son of the first Duke of St. Albans (ib. xxvi. 91).
On 14 Nov. 1757 Draper, still a lieutenant and captain 1st foot guards, was commissioned as lieutenant-colonel commandant to raise a regiment of foot a thousand strong for service in the East Indies. The regiment took rank as the 79th foot, but in an early impression of the army list for 1758 figures wrongly as the 64th. The rendezvous was at Colchester. The regiment was partly formed of companies drafted entire from the 4th, 8th, and 24th foot, and the authorities appear to have considered the old-fashioned wooden ramrods good enough for it, in place of steel (see War Office Marching Books and Warrant Books, under date). Draper arrived at Madras with the regiment, which lost fifty men by ‘Brest fever’ (ship-typhus) on the way out, in the Pitt Indiaman on 14 Sept. 1758 (Orme, ii. 368), and at its head repeatedly distinguished himself during the siege of Fort St. George from November 1758 to January 1759 (ib. pp. 390–459). When Stringer Lawrence resigned on account of ill-health in February 1759, the command of the troops in Madras devolved on Draper, who was too ill to take it up, and returned home soon afterwards (ib. ii. 463). Early in 1760 Draper was appointed deputy quartermaster-general of a projected secret expedition under Major-general Kingsley (Home Off. Mil. Entry Book, xxvi. 5). The expedition was originally intended to proceed to Mauritius and Bourbon (Réunion), but this was changed, and it was secretly instructed to rendezvous at Quiberon for an attack on the fortress of Belle Isle, on the coast of Brittany. Various circumstances, including the death of the king, delayed the operations, and on 13 Dec. 1760 the authorities, as the season was so far advanced, ordered the troops, which had been long on board ship at Spithead, to be relanded (Beatson, Nav. and Mil. Memoirs, ii. 420, iii. 167 n.) Draper held no rank in the expedition which captured Belle Isle the year after. He was promoted colonel 19 Feb. 1762, and in June that year again arrived at Madras with the rank of brigadier-general, in the Argo frigate, to assume command of an expedition against Manilla. His original instructions are preserved among Lord Leconfield's manuscripts, and are printed at length in ‘Hist. MSS. Comm.’ 7th Rep. 316 et seq. Under Draper and Admiral Cornish the expedition appeared off Manilla unexpectedly 25 Sept. 1762. A landing was effected with great difficulty owing to the advanced season, and on 6 Oct. 1762 the place was carried by assault with comparatively little opposition, the victors accepting bills on Madrid for a million sterling in lieu of pillage (Beatson, ii. 496–515, iii. 185 n.) Draper returned home at once and presented the Spanish standards to his old college. On Wednesday, 4 May 1763, ‘the Spanish standards taken at Manilla by General Draper, late fellow, were carried in procession to King's College chapel by the scholars of the college. A Te Deum was sung, and the Rev. W. Barford, fellow and public orator, delivered a Latin oration. The flags were placed on either side of the altar-rails, but were afterwards removed to the organ-screen’ ({sc|Cooper}}, Annals of Cambridge, iv. 327). The state of affairs at Manilla after Draper's departure is detailed in ‘Calendar Home Off. Papers,’ 1760–5, pp. 584–9. The Spanish court refusing to recognise the treaty, Draper strongly urged the government to insist on payment of the ransom, his share of which amounted to 25,000l. He published his views in a pamphlet entitled ‘Colonel Draper's Answer to the Spanish Arguments claiming the Galleon and refusing Payment of the Manilla Ransom from Pillage and Destruction’ (London, 1764). But the government were not in a position to press the matter, and Draper, recognising the hopelessness of the case, let it drop. He had been in 1761 appointed governor of Great Yarmouth, a post worth 150l. a year, and on 13 March 1765 he was made colonel of the 16th foot, his old corps, the 79th, having ceased to exist. On 4 March 1766 he received
