Most of the hits on the first page of a Google search on ‘first British casualty of wwi’ state that it was Private John Parr of the 4th Battalion of the Middlesex Regiment, who was killed on 21 August 1914. The Daily Mail reports that Henry Hadley, a languages teacher working in Berlin in 1914, was shot by a German soldier on 3 August after an argument on a train that was taking him out of Germany. He died at 3:15 am German time on 5 August, three and a quarter hours after Britain declared war.
Only the Great War Forum states, correctly, that the first British servicemen to by killed by enemy action in the First World War were members of the crew of HMS Amphion, which struck a mine and sank at 6:30 am on 6 August with the loss of about 150 men. The poster Crooneart says that the first dead are considered to be Stokers 1st Class Jesse Foster and Albert Martin, Stoker 2nd Class William Dick and Leading Stoker Henry Copland, who are all given as dying at 6:30 am on 6 August. Presumably they were in the part of the ship closest to the explosion and were killed instantly.
To be fair to Wikipedia, its entry on Parr does link to its entries on Amphion and Hadley, but the other sites produced by the search appear to assume that first British casualty means first British soldier killed, ignoring naval casualties.
At dawn on 5 August, the Harwich Force of two destroyer flotillas, supported by armoured cruisers and submarines, sailed on a sweep towards the Dutch coast. Captain H. C Fox, captain of Amphion, was in command of the 3rd Destroyer Flotilla. At 10:15 am a British trawler informed the destroyer HMS Laurel that it had seen a ship ‘throwing things overboard’
Correctly assuming that the ‘things’ were mines, Fox ordered his flotilla to spread out and search for the suspicious ship. The destroyers HMS Lance and Landrail sped ahead to the point at which the trawler had seen her. Around 11 am they spotted the German minelayer Königin Luise, which in peacetime ferried passengers between Hamburg and Heligoland. She was painted in the colours of a British steamer of the Great Eastern Railway.
Lance and Landrail, supported by Amphion, gave chase and sank Königin Luise by noon. This was the first ship to be sunk in the war and the first casualties to be inflicted on the enemy by British. The shots fired were not, however, the first ones of the naval war; the German Mediterranean squadron of the battlecruiser SMS Goeben and the light cruiser SMS Breslau had bombarded the French ports of Philippeville and Bône in Algeria on the evening of 3 August.
On the way back to Harwich Amphion spotted another steamer in Great Eastern Railway colours, this one flying a German flag. The flotilla opened fire, but the steamer then raised the British Red Ensign. She was the Great Eastern Railway steamer St Petersburg and was carrying the German ambassador, his wife and staff to the neutral Netherlands on their way back to Germany. The destroyers initially ignored Fox’s orders to cease fire, and he had to place Amphion between them and the St Petersburg in order to stop them firing.
At 6:30 am on 6 August Amphion struck one of Königin Luise’s mines. The crew were ordered to abandon ship, but she hit another mine almost immediately and sank quickly. As well as about 150 of her own men, 18 survivors of Königin Luise went down with Amphion.
 Quoted in J. S. Corbett, H. Newbolt, Naval Operations, 5 vols. (London: HMSO, 1938) vol. i, p. 38.