George Pocock was born in 1706, the son of Thomas Pocock FRS, Chaplain to the Royal Navy at Greenwich Hospital and Rector of Danbury in Essex, by his wife, Joyce, a daughter of James Master of East Langdon in Kent, and sister of Margaret, wife of George Byng, Viscount Torrington. The Pocock family had been resident at Chieveley in Berkshire for at least the previous two hundred years. Thomas was a grandson of Dr. Laurence Pocock, the Rector of the adjoining village of Brightwalton and a 2nd cousin of Prof. Edward Pocock, the great Middle Eastern linguist. In 1718, George entered the Navy under the charge of his uncle, Streynsham Master, on board the Superbe, in which he was present at the Battle of Cape Passaro. He was, afterwards for three years, on board the Looe, with Captain George Prothero, for a year in the Prince Frederick, and another in the Argyle; and passed his examination on 19 April 1725.

From 7th December 1726 to May 1728, he was lieutenant of the Burford, with the Hon. Charles Stewart. Afterwards, he served on board the Romney, with Charles Brown; on the Canterbury, with Edmund Hook; in the fleet in the Mediterranean, under Sir Charles Wager; and on the Namur, carrying Wager's flag. On 26th February 1734, he was promoted to be commander of the Bridgwater fireship. On 1st August 1738, he was posted to the Aldborough frigate, attached to the fleet in the Mediterranean under Rear-Admiral Nicholas Haddock. The Aldborough was paid off at Deptford in December 1741 and, early in the following year, Pocock was appointed to the Woolwich of 40 guns, which he commanded in the Channel during the year. In January 1743, he was moved onto the 80-gun ship Shrewsbury, much against his will, the smaller ship being, he considered, more advantageous in time of war. During the few weeks he was aboard the Shrewsbury, he occupied himself in pointing out her defects in writing to his cousin, Lord Torrington, and complained of being moved, against his will, into a large ship. His interest prevailed. He was appointed to the Sutherland, of 50 guns, and sent for a cruise in the Bay of Biscay and on the north coast of Spain.

In 1744, Pocock convoyed the African trade to Cape Coast Castle and brought home the East India ships from St. Helena. In 1745, he again took out the African trade and, crossing over to the West Indies, joined Commodore Fitzroy Henry Lee with whom, and afterwards with Commodore Edward Legge, he continued on the Leeward Islands station. Upon Legge's death, on 18th September 1747, he succeeded to the chief command. Shortly afterwards, a letter from Sir Edward (afterwards Lord) Hawke giving him the news of the victory over L'Etenduere on 14th October, warned him to look out for the convoy which had escaped. This he did with such good effect that about thirty of the ships fell into his hands, and some ten more were picked up by the privateers. Early in May 1748, he was relieved by Rear-Admiral Henry Osborne and returned to England in the following August. For the next four years, he resided in St. James' Street, London and, in July 1752, was appointed to the Cumberland on the home station. In January 1754, he commissioned the Eagle and, in March, sailed for the East Indies with the squadron under the command of Rear-Admiral Charles Watson. The squadron put into Kinsale where, in a violent gale, the Eagle parted her cables, fell on board the Bristol and was only saved from going on shore by cutting away her masts. The two ships were consequently left behind when the squadron sailed and Pocock was ordered to take them to Plymouth to refit. He was not able to reach Plymouth until 15th April and, a few days later, he and his ship's company were turned over to the Cumberland, in which he went out to the East Indies.

On 4th February 1755, he was promoted to be rear-admiral of the white and, hoisting his flag on board the Cumberland, remained with Watson as second in command. On 8th December 1756, he was advanced to the rank of vice-admiral and, on Watson's death on 16th August 1757, succeeded to the chief command. At Madras, in March 1758, he was joined by Commodore Charles Steevons and, having moved his flag to the Yarmouth of 64 guns, he put to sea on 17th April, his squadron now consisting of seven small ships of the line, ships of 64, 60 or 50 guns. On the 29th, off Fort St. David, he fell in with the French squadron of about the same nominal force, all being ships of the French East India company, except the one 74-gun ship which carried the broad-pennant of the Comte d'Ache. Pocock led the attack as prescribed by the English 'Fighting Instructions'. An indecisive action followed, the French practising the familiar manoeuvre of withdrawing in succession and reforming their line to the leeward side. Battles fought in this manner never led to any satisfactory result. It generally happened that some of the English ships were unable to get into action in time and, on this occasion, as on many others, the captains of the rearmost ships were accused of misconduct. Three were tried by court-martial, found guilty of not using all possible means to bring their ships into action, and severally sentenced to be dismissed from the ship, to lose one year's seniority, and to be cashiered. The court failed to recognise that the manoeuvre required of them was practically impossible.

On 1st August, the two squadrons were again in sight of each other off Tranquebar, the French, with two 74-gun ships, having a considerable nominal superiority. It was not, however, till 3rd that Pocock succeeded in bringing them to action, and then, in the same manner and with the same indecisive result. The French then went to Mauritius and Pocock, having wintered at Bombay, returned to the Coromandel Coast in the following spring. The French fleet of eleven ships did not come to the coast until the end of August and, on 2nd September, it was sighted by the English. After losing it in a fog and finding it again, on the 8th, off Pondicherry, Pocock brought it to action two days later, but again in the manner prescribed by the 'Fighting Instructions’, and with unsatisfactory results. The fighting was more severe than in the previous actions. On both sides, many men were killed and wounded, and the ships were much shattered, but no advantage was gained by either party. That the prize of victory finally remained with the English was due, not to Pocock and the East Indian squadron, but to the course of the war in European waters. In the following year, Pocock returned to England, arriving in the Downs on 22nd September. On 6th May 1761, he was nominated a Knight of the Bath and, about the same time, was promoted to be admiral of the blue.

In February 1762, Pocock was appointed commander-in-chief of 'a secret expedition' destined, in fact, for the reduction of Havana, which sailed from Spithead on 5th March, the land forces being under the command of the Earl of Albemarle. On 26th April, it arrived at Martinique, but sailed again on 6th May. Taking the shorter though more dangerous route on the north side of Cuba under the efficient pilotage of Captain John Elphinston, Albemarle and the troops landed six miles to the eastward of Havana on 7th June, under the immediate conduct of Commodore Keppel, Albemarle's brother. The siege-works were at once commenced. A large body of seamen were put on shore and "were extremely useful in landing the cannon and ordnance stores of all kinds, manning the batteries, making fascines, and in supplying the army with water". By the 30th, the batteries were ready and, on 1st July, opened a heavy fire, supported by three ships of the line, under the immediate command of Captain Hervey of the Dragon. The Moro was engaged, but, after some, six hours, the ships were obliged to haul out of action, two of them - the Cambridge and the Dragon - having sustained heavy loss and much damage. After this, the work of the fleet was mainly limited to preventing any movement on the part of the Spanish ships, which might otherwise have effectually hindered the English works. The English batteries gradually subdued the enemy's fire, though the Spaniards were materially assisted by the climate, which rendered the exposure and fatigue very deadly. By 3rd July, more than half of the army and some three thousand seamen were down with sickness. Under all difficulties, however, the siege was persevered with. The Moro was taken by storm on 30th July and, on 13th August, the town, with all its dependencies and the men-of-war in the harbour - to the number of twelve ships of the line, besides smaller vessels - surrendered by capitulation. The monetary value of the prize was enormous. The share of Pocock alone, as Naval Commander-in-chief, was £122,697. 10s. 6d; that of Albemarle was the same. In November, Pocock delivered over the command to Keppel, who had just been promoted to flag rank, and sailed for England with five ships of the line, several of the prizes, and some fifty of the transports. The voyage was an unfortunate one. Two of the line-of-battle ships, worn out and rotten, foundered in the open sea, though happily without loss of life. Two others had to throw all their guns overboard and, with great difficulty, reached Kinsale. Twelve of the transports went down in a gale, many were wrecked in the Channel, with the loss of most of their crews and, in those ships which eventually got safe in, a large proportion of the men died, worn out with fatigue, hunger, thirst and cold. Pocock, in the Namur, arrived at Spithead on 13th January 1763.

Pocock had no further service and, in a letter to the Admiralty, dated 11th September 1766, stated that "the King had been pleased to grant his request of resigning his flag," and desired that "his name might be struck off the list of admirals," which was accordingly done. It was generally believed that this was in disgust at the appointment of Sir Charles Saunders, his junior, to be First Lord of the Admiralty. Although Saunders' patent, which was dated 15th September, may have been the deciding reason, the prospect of continued peace, his large fortune and a wish not to stand in the way of his poorer friends doubtless had their weight. Pocock had married, in November 1763, Sophia Pitt, daughter of George Francis Drake, grandaughter of Sir Francis Drake of Buckland Monachorum in Devon, third baronet, and widow of Commodore Digby Dent. By her, he left issue a daughter and one son, George (1765-1840), created a baronet at the coronation of George IV. They lived at Chieveley in Berkshire and in Mayfair, where Sir George died at his house in Curzon Street on 3rd April 1792. He was buried at Twickenham, but there is a monument to his memory is in Westminster Abbey.