Cockburn joined the Royal Navy in 1926 and became a Midshipman on 15 September 1926.

He became Sub Lieutenant on 28 April 1930.

He was attached to the RAF in January 1931, when he received a temporary commission in the RAF as a Flying Officer on attachment on 18 January 1931.
At the same time, he had been promoted to Lieutenant in the Navy.

He became a Flight Lieutenant on 1 July 1935 and relinquished his RAF commission on 26 July 1939.

On 1 March 1939 he was promoted to Lieutenant Commander in the Navy

Cockburn commanded 718 (Catapult) Squadron from 29 March 1939.

Luftwaffe attacks on the Fleet anchorage at Scapa Flow led to the Royal Navy forming 804 Squadron.
On 25 November 1939 four Sea Gladiators of 769 Squadron were detached to Hatston on Orkney and, on 30 November, the detachment became 804 Squadron.
The Squadron was at this time under Fighter Command’s control and remained so during the Battle of Britain period.
Cockburn took command of 804 Squadron on 9 December.

During the evacuation from Norway, 804 Squadron operated from HMS Glorious.

On 1 May, during the evacuation of the first Norwegian campaign, HMS Glorious operated together with HMS Ark Royal off the Norwegian coast. During the early morning at about 07:00, two of 802 Squadron's Sea Gladiators intercepted a lone Do 17P of 1(F)./120, which had just discovered Vice-Admiral Well's force. The Dornier was left in a damaged condition but the alarm had been raised. A force of Ju 87Rs from I./StG 1 took off from their base in Norway, led by two He 115s of 2./KüFlGr. Arriving over the naval force, one group of Stukas bombed HMS Glorious but failed to achieve any hits. The other group were driven off by a section of Sea Gladiators.
At 15:00, Blue Section from 804 Squadron (Lieutenant Richard Smeeton, Acting Sub Lieutenant R. R. Lamb and Lieutenant Taylor) engaged a He 111 at 16,000ft. Two of the pilots were able to get in a short burst each before the Heinkel escaped. 20 minutes later another Heinkel was spotted but the Sea Gladiators could not get close enough to open fire.
At 16:30, Red Section from 804 Squadron (Lieutenant R. H. P. Carver, Lieutenant C. E. F. Gibson and Sub Lieutenant Michael Fell) took off to relieve Blue Section. Not long after take-off a lone raider was seen at low altitude. This was chased off and the section climbed up to 18,000ft. Not long after they were ordered to intercept another lone aircraft, again at low altitude. A He 115 was observed low over the water and despite a long chase, in which all their ammunition was expended, the Heinkel escaped.
At this point an enemy raid was detected on radar and further sea Gladiators were scrambled from HMS Glorious to intercept. Blue Section from 802 Squadron scrambled two Sea Gladiators at 18:00 and these were followed 15 minutes later by a section from 804 Squadron. The 802 Squadron pair, led by Lieutenant J. F. Marmont, went after another lone raider, possibly a He 115, but this escaped. By this time the 804 Squadron section had become engaged with a force of six Ju 87s. Lieutenant Marmont had also joined the fray and getting in behind one of the Stukas, Marmont shot it down into the sea west of Namsos at 18:25. The crew of Oberfeldwebel Erich Stahl and Unteroffizier Friedrich Gott from 2./StG 1 were rescued by a Royal Navy destroyer.
Lieutenant Commander Cockburn of 804 Squadron took part in the attack on the remaining Ju 87s and reported:

"At 1825 hrs, six Ju 87 dive-bombers were sighted three miles ahead on an opposite course in open "V" formation. The order was given to open fire, and the section half-rolled individually onto the tails of the aircraft, each pilot attacking one enemy. Fire was maintained in short bursts, as the enemy twisted and turned, until the final bombing dive was commenced. The attack was broken off at this point, as I imagined, quite erroneously, that the pom-pom fire would take effect below this."

In November 1940 he left 804 Squadron.

On 1 June 1941 Cockburn took command of 881 Squadron and in 1942 was commander (Flying) on HMS Argus.

In 1943 he held the same appointment on HMS Stalker, which was part of Force V in the assault on Salerno.

He led 26 Seafires of the Naval Fighter Wing to the airfield at Paesturn on 9 September 1943, to operate from there until the RAF squadrons arrived in Sicily.
The Wing then moved to Asa to operate alongside 324 Wing of the RAF, which it did for six days before returning to its own ships.
Cockburn was awarded the DSC on 23 May 1944 for his part in the action.

From December 1944, Cockbunr commanded the Royal Navy Air Station at Puttalam, Ceylon.

Cockburn ended the war with 1 biplane victory.

He continued to serve after the war and was promoted to Commander on 31 December 1946.

On 31 December 1950, he was promoted to Captain.

In 1955, he was Naval Attaché, Rio.

He retired as in 1960.

Claims:

Kill no.	Date	Time	Number	Type	Result	Plane type	Serial no.	Locality	Unit
	1940
	01/05/40	18:25	1	Ju 87 (a)	Damaged	Sea Gladiator	N2276/H	W Namsos	804 Squadron

Biplane victories: 1 damaged.
TOTAL: 1 damaged.
(a) Possibly claimed in combat with Ju 87s from 2./StG 1, which suffered at least one lost Ju 87R.

Sources:
Fledgling Eagles - Christopher Shores with John Foreman, Christian-Jaques Ehrengardt, Heinrich Weiss and Bjørn Olsen, 1991 Grub Street, London, ISBN 0-948817-42-9
Flying Sailors at War: Volume 1 – Brian Cull with Bruce Lander and Mark Horan, 2011 Dalrymple & Verdun Publishing, Stamford, ISBN 978-1-905414-14-7
Gladiator - Andrew Thomas, 2003 Battle of Britain, Key Publishing Ltd, Stamford
Gloster Gladiator Aces - Andrew Thomas, 2002 Osprey Publishing, London, ISBN 1-84176-289-X
Gloster Gladiator Home Page - Alexander Crawford
Gloster Gladiator vol. 1 Development and Operational History - Alex Crawford, 2009 MMP, ISBN 978-83-89450-59-3
Men of the Battle of Britain - Kenneth G. Wynn, 1999 CCB Associates, ISBN 1-902074-10-6
Royal Navy Aces of World War 2 - Andrew Thomas, 2007 Osprey Publishing, Oxford, ISBN 978-1-84603-178-6
Those Other Eagles – Christopher Shores, 2004 Grub Street, London, ISBN 1-904010-88-1