Oh hey, it's the question Stalin and the Soviet government were asking until 1944!

Anyway, I'll try to make this a non-list answer...but it might come across that way, and I need to link to answers for further justice to many of these topics. The Blitz is traditionally given as ending in May 1941, although it's not like this was a formal end. It was the last major attack by the Luftwaffe on London before resources began to be repositioned for the impending attack on the USSR. The Luftwaffe would return for a series of air attacks on the UK over the course of the war, such as in the "Baedeker" raids in March 1942, and Operation Steinbock/the "Baby Blitz" in early 1944. These were replaced with attacks by V-1 missiles in June 1944 and V-2 missiles in September 1944, and those attacks continued all the way to late March 1945. While all of these raids caused much less destruction and strategic danger to Britain, they did mean maintaining active air defenses over Britain and dealing with civilian casualties all the way almost to the end of the war.

Britain's own air war was a major undertaking, and as I discuss in that linked answer that I wrote British (and later American) strategic bombing of Germany was seen initially a means of giving Germany a taste of its own medicine, and then a substitute for a Second Front. The RAF had been bombing German targets from mid-1940 on, and continued to do so throughout the war, although bombings in 1940 and 1941 were extremely limited in geographic area and effectiveness. The strategic bombing campaign began to adopt "dehousing" and area bombing as a strategy in 1942, and from June 1943 the Combined Bomber Offensive got underway, with the RAF focusing on night time area bombings (ie, burning down German cities) while the USAAF focused on daylight raids on infrastructure (in theory, although the results were often similar to the RAF). While Dresden at the end of the war gets all of the controversy, there were major raids over the course of the war, such as Hamburg being targeted in a bombing raid in July 1943 that caused a firestorm killing some 30,000-40,000 people, and Berlin being targeted in major air offensives in September 1943 and February 1944.

As for the naval war - a major campaign over the course of the entire war was the Battle of the Atlantic, the attempt by German submarines to enforce a naval blockade on the British Isles and starve it of resources. The major part of the campaign was fought between mid-1941 and mid-1943, when German U-boats were a major threat to transatlantic convoys. I honestly won't be able to do this subject justice, so for more information I would recommend this answer by u/thefourthmaninaboat. I would also note that the Royal Navy was also dealing with maintaining supply routes and convoys to the USSR after June 1941, and this involved a lot of action against German forces along the Northern Route to Murmansk.

A major focus of the British army from 1941 to 1944 was in the Mediterranean. This was part of the "Peripheral Strategy", where rather than risk a direct attack on German-held continental Europe, British forces would operate against German and Italian forces in the Mediterranean and undermine the Axis that way.

In the spring of 1941, this involved supporting Greece in its defense against Axis invaders. British and Commonwealth forces had been stationed in Greece from March, and the Axis launched a swift and successful invasion of the country about a month later. Probably the most notable action in that campaign was the Battle of Crete, where German paratroops captured the island in a pyrrhic victory. u/fourthmaninaboat as some background on that battle here and u/Georgy_K_Zhukov has some information on the consequences of the battle here.

Perhaps the most notable campaign in this theater was the Western Desert campaign, where the British 8th Army (especially under General Montgomery, who commanded it from mid 1942 to mid 1943) engaged the Italian army and German Afrika Korps (famously commanded by Erwin Rommel) in Libya and Egypt. The most notable battles were the sieges of Tobruk (successfully defended by British and Commonwealth forces in 1941, captured by the Axis in 1942) and the Battles of El Alamein in 1942, the second one of which, fought in October/November 1942, gets treated as a decisive turning point in the war, comparable to the Battle of Midway or the Battle of Stalingrad (Churchill later wrote: "Before Alamein we never had a victory. After Alamein, we never had a defeat.").

The British and Americans formed a Combined Chiefs of Staff from mid 1942 on to coordinate strategy, and this resulted in a number of combined campaigns. Besides the afore-mentioned Combined Bomber Offensive, it included Operation Torch and the North African campaign from November 1942 to May 1943, which saw the complete defeat of Axis forces on the continent, followed by the invasion of Sicily and Italian campaign from July 1943 to the end of the war. British and Commonwealth forces were involved in significant battles such as the Landings at Salerno and the Battle of Monte Cassino. A further discussion of the Italian Campaign and Churchill's focus on the "Soft Underbelly of Europe" can be found here, starting with an answer by u/Ornlu_Wolfjarl.

A final point should be made that British and Commonwealth forces weren't just engaged strictly around Europe in 1941-1944, but across the world. Italian East Africa was conquered in a campaign in 1941, the British conducted a series of campaigns against Vichy colonies, such as Madagascar and Syria, and also had to contend with war against the Japanese from December 1941. The battleships Prince of Wales and Repulse were sunk by the Japanese in a Pearl-Harbor-for-the-British on December 10, 1941 (discussed here by DBHT14...I think I'm out of mentions for this comment), and the British lost major land offensives to the Japanese resulting in the surrender of Hong Kong and Singapore in late 1941-early 1942. By mid 1942, British focus in the region was on fighing along the Burma-India frontier. Notable actions in Burma included the Arakan Offensive in 1942, the Chindit Operations behind Japanese lines over the course of 1942-43, and the Japanese invasion of India in the spring of 1944, culminating in Japanese defeats at the Battles of Imphal and Kohima.

And all of this while the British were extremely busily engaged in constant diplomacy, especially with their US and Soviet counterparts, as well as preparing for Operation Overlord in June 1944. This is by no means an exhaustive description of British operations and actions in this time period. They were incredibly busy across the globe.