While the USSR only became directly involved in the Pacific Theatre in August 1945, the British Empire was involved from the very start of the fighting in the Pacific. British and Indian forces fought in Hong Kong, Malaya and Burma, while the Australians and New Zealanders played an important part in the fighting in Indonesia, New Guinea and the Solomon Islands. The Royal Navy, and its various subsidiaries from the Empire, fought the Imperial Japanese Navy in the Indian and South Pacific, sometimes in companionship with the United States Navy, sometimes alone. In this answer, I will focus mainly on the naval side of the fighting - I'm not as well versed in the fighting in Burma and elsewhere - this answer by /u/Bernardito covers the Burma side of things well

In the 1920s and early 1930s, Japan was seen as the main threat by the Royal Navy. With its strong fleet, it posed a major threat to British interests in the Far East, especially Singapore, Malaya and Sarawak (on the north coast of Borneo), as well as the major naval base at Hong Kong. Japanese aggression in China only raised these fears. To counter these, the RN began to build up Singapore as a better-protected naval base than Hong Kong. This was in support of its main strategy for a war in the Pacific, War Memorandum (Eastern) - analogous to the American War Plan Orange. War Memorandum (Eastern) called for the ships of the China Station to hold off the Japanese in the 'Malay Barrier' (the gap between Vietnam and the Philippine island of Palawan), while the main battlefleet deployed from the UK to Singapore. En route, it would refuel at a number of secret bases - Base M (Kamaran Bay in the Red Sea), Base T (Addu Atoll) and Base W (Nancowry Harbour). Once the fleet reached Singapore, it was to proceed in three phases - firstly, relieving Hong Kong, which was assumed would be easily cut-off in time of war. Secondly, the fleet would establish a forward base in the Ryuku Islands, from which it could operate against Japanese trade, forcing the IJN to respond to it. Finally, it would use these attacks on trade to bring the IJN's battlefleet to a decisive action. While this plan was well-worked out, it never really got a chance to be used. In the mid-30s, European threats started to emerge, as Germany and Italy started to adopt a more belligerent foreign policy. The RN did not have enough strength to face both the IJN and the navies of the European powers, and so focused on those closer to home. Continued Japanese attacks on China also helped to limit the extent to which the plan could be implemented, putting Japanese troops on the border with Hong Kong. While there were plans for the RN to build up to a 'New Standard' in the late 1930s, which was intended to give the RN the capabilities to defeat both the German and Japanese navies, the German invasion of Poland put paid to this. Ships needed to face the Japanese were cancelled or delayed to build the escorts and merchants needed to replace losses in the Battle of the Atlantic, while ships from the China Station and the Eastern Fleet were diverted to the Atlantic or Mediterranean. Monetary constraints in the 1920s and 1930s meant that many of the RN's older ships were unmodernised, while most of their IJN counterparts had been. All of this meant that the RN was in a poor state to respond to continued Japanese aggression in 1941.

This continued aggression was highly alarming to the Admiralty. In particular, Japan's seizure of French Indochina (including the naval base at Cam Ranh Bay, which the RN had hoped to use) was a clear threat to Britain's interests, putting Singapore and Malaya within range of Japanese aircraft based there. In the months following the seizure, plans to build up the Eastern Fleet to significant strength were developed; these called for a strength of seven (mostly older) capital ships, an aircraft carrier, ten cruisers and 24 destroyers. It was expected that this force, in conjunction with the strong American Pacific Fleet, would deter further Japanese moves. However, it was not thought that this could be put in place before March 1942, thanks to the need to repair and refit ships, and to put together enough of the supporting vessels. Churchill, who was convinced that a smaller force could still be effective. He pushed for a force consisting of one modern battleship, one of either of the battlecruisers Repulse and Renown, an aircraft carrier, and escorts. Ultimately, Churchill would get his way, as the RN encountered somewhat of a crisis in October-December 1941. Three of the battleships of Mediterranean Fleet were heavily damaged or sunk, along with the carrier Ark Royal. This, combined with the many ships damaged during the earlier part of the year, left the fleet heavily outnumbered by the Italians; ships for the Far East were difficult to find. Political considerations were also present, with Churchill's more prestigious force also intended to act as a signal to Britain's allies and colonies in the region that Britain saw defending them as a serious matter. Churchill's smaller fleet, given the name Force Z, and consisting of Prince of Wales and Repulse, arrived at Singapore on the 2nd December 1941. It may have been intended that the new carrier Indomitable would join them, but she had run aground at Kingston while working up, and was under repair.

Force Z had a few short days at Singapore before the Japanese attacked. Its commander, Admiral Tom Phillips, spent most of this time conferring with the American commanders in the Philippines, and with the Australians, in order to better arrange defences in case of war, and to gain reinforcements for his force. Just after midnight on the 7th, the first Japanese landings in Malaya occurred at Kota Bharu, in the north of the peninsula. The Japanese were soon able to establish a beachhead, supported by troops landed in Thailand, which had a government friendly to the Japanese. At dusk on the 8th Phillips took Force Z north to raid the invasion convoys at the Thai port of Singora. He avoided a surface action with the Japanese covering force on the 9th, as he turned to investigate reports of further landings at Kota Bharu, but when these proved illusory and his force was sighted by a Japanese aircraft, began to return to Singapore. On the 10th, false reports of landings at Kuantan reached Phillips. While travelling to investigate them, Force Z was sighted by submarines and aircraft; while Phillips turned back after his own scout aircraft reported Kuantan was clear, this did not take him out of range of Japanese attack aircraft. Force Z was attacked by several waves of aircraft, launching torpedo and bomb attacks. Prince of Wales was immobilised quickly, having taken a highly damaging torpedo hit to the stern early in the engagement, before taking three more hits. Repulse managed to avoid several torpedoes, but was unable to avoid every one. She took four hits from a well-executed attack, and sank at 12:30, about 90 minutes after the first attack. Prince of Wales remained afloat for 45 more minutes. The Japanese lost just three aircraft.

With these two losses, the most powerful Allied surface force in the Far East was the scratch force of what would become, on the 1st January 1942, ABDACOM, the American-British-Dutch-Australian Command. This force was a scratch force of cruisers and destroyers, mostly older, obsolete ships, and was tasked with protecting Malaya and the Dutch East Indies. The American and Dutch elements fought a number of raiding actions against the Japanese, some successfully, many more unsuccessfully. On the 10th February, two British destroyers made an attempt to raid Japanese shipping off Endau on the Malay peninsula, but did little damage, at the cost of HMS Thanet. Singapore fell on the 15th, unhinging the Allied position in Indonesia, and allowing the Japanese to begin landing on Sumatra. This in turn threatened the main ABDA naval base, at Surabaya on Java. The ABDA force attempted to raid the Japanese landings at Palembang, but were forced to turn back in the face of strong air attacks. On the 19th, Japanese land-based aircraft raided Surabaya, while carrier aircraft struck the port of Darwin, the main ABDA logistics base. Little damage was done at Surabaya, but many ships were sunk in Darwin harbour, along with many of the port installations. These were merely preliminaries for the Battle of the Java Sea, the final destruction of the ABDA force. It was clear that Java was the next target for a Japanese attack, which came on the 27th. The ABDA force sortied to meet the invasion convoys, but was engaged and scattered by a better organised and commanded IJN force. The British destroyers Electra and Jupiter were sunk, along with two Dutch cruisers and a Dutch destroyer, while the heavy cruiser Exeter was heavily damaged. Over the next few days, the remainder of the force was hunted down and sunk; Perth and Houston were sunk in the Sunda Strait on the 28th, while on the 1st March, Exeter and her escorts Encounter and the American Pope were sunk in the Java Sea. With these losses, the Japanese were free to secure the Dutch East Indies, putting them into a position to attack Australia, or to move into the Indian Ocean.

While it never quite met Churchill's ambition of contributing 1/3 of its resources against Japan, Britain was quite heavily involved in the Pacific theatre, though moreso in the Indian Ocean theatre until the final months, it is still regarded as playing a part of the Pacific War. It's often overlooked in most European histories, even British histories, due to the 'holding action' role Britain took, and the lack of any significant offensives until much later in the war.

British forces in the Far East were generally poorly equipped and trained, since most of Britain's crucial war materiel was tied down in Africa, the Middle East and the British Isles themselves. They lacked the large warships of the Home and Mediterranean fleets, and crucially any carriers with the exception of the venerable Hermes; ground forces lacked armour and air force units were equipped with more obsolescent hurricanes and almost no radar. As a result, they took heavy losses in late 1941 and 1942, losing Penang, Hong Kong, Malaya, Singapore, the Solomons, and Brunei. ABDACOM (Australian-British-Dutch-American-COMmand) was headed by British General Archibald Wavell from January 1, 1942, though the collapse of so many ABDA units forced his resignation in February. The Japanese sinking of the HMS Repulse and Prince of Wales, the Battle of the Java Sea and the subsequent Indian Ocean Raid forced the overstretched Royal Navy to retreat to the Western side of the Indian Ocean, and allowed Japanese naval units to operate effectively unchecked in the Bay of Bengal and the South West Pacific.

The loss of naval supremacy prevented significant resistance to the Japanese invasion of Burma in 1942, which would prove Britain's most significant contribution against Japan, the Western Allies' only continuous land campaign in the Pacific. Britain's longest retreat eventually halted and consolidated on India's eastern border as the Monsoon season broke. In accordance with the 'Europe First' Policy operations in Burma and the Pacific were exclusively a holding action until the defeat of Germany, as a result, the only offensive operations in 1943 were long-range penetration raids by 'Chindits' and 'Merril's Marauders', and a failed thrust along Burma's Arakan coast. British submarines returned to the Indian Ocean and to the Pacific however, many operating from Ceylon and Australia, sinking some 40,000 tons of Japanese shipping. RAF transport aircraft cooperated with USAAF craft in flying 'the Hump', supplying Chinese Nationalist forces from bases in India by flying over the Himalayas, and some British-led Solomon Island coast watchers also assisted US marine efforts in taking Guadalcanal in 1943. SOE also set up Task Force 136, to facilitate anti-Japanese resistance across South East Asia.

In March 1944, British involvement in the Burma campaign stepped up following the commencement of Japan's U-Go offensive against India. Under the command of General William Slim, Japanese forces were stopped cold in their advances against Kohima and Imphal, becoming cut off from supply lines and gradually starving. The eventual retreat saw the British return to the offensive in Burma, and the Japanese lose some 60,000 men, it's largest defeat up to that point. Though monsoons slowed the advance, by early 1945 British and Indian troops had retaken Meiktila and Mandalay, reaching central Burma. The capital at Rangoon was recaptured by April, and the Japanese Burma Area Army was effectively destroyed at Sittang in August. Perhaps just as significant, the Royal Navy also returned to the Pacific War, striking Japanese targets in Sumatra and Java in Operation Meridian in January 1945. This was the newly formed British Pacific Fleet, with 21 carriers of various type, 4 battleships, 11 cruisers, 35 destroyers and 750 aircraft. the RNs largest ever. Though really more of a political effort than a military one, the BPF supported the US Fifth Fleet as Task Force 57 during the invasion of Okinawa. British carriers were notable for their armoured decks, unlike their US counterparts, when some, such as HMS Victorious suffered several Kamikaze attacks to little effect. Strikes were also carried out against the Japanese island of Honshu, and the BPF and Commonwealth infantry would certainly have taken part in an Invasion of Japan. Halsey and King, however, were reluctant to use the BPF due to the potential post-war claims Britain could make if it played any decisive role.

​Sources used;

Roskill, S.W. 1960. The Navy at War 1939-1945.

Grehan & Mace. 2015. The Battle for Burma 1943-1945.

Farrington, K. 2005. Victory in the Pacific.

The USSR was neutral until the last days of the conflict. For a bit more on Soviet Forces in the Far east, I'd point you here.