Home defence requires the UK to have sufficient mastery of the Channel and neighbouring seas, and of our airspace, to make invasion impossible or unacceptably costly to any potential enemy. We would normally expect NATO support, but having our own forces in place for any sudden initial attack remains vital.
The UK successfully prevented invasion by the Spanish in 1588, and by the French in the Napoleonic wars. These were achieved by sea power. The resistance to German invasion in the last century required air power and sea power, which were deployed successfully. We nonetheless experienced some shelling and bombing at home in the first world war, and major bomb attacks in the Second World War. The airforce had to deal with fighter and bomber incursions on a grand scale, and to combat the development of missile technology with German flying bombs and rockets at the end of the conflict.
Today we therefore need sufficient sea and air power to act as a deterrent to any potential aggressor. We also need the industrial capability to scale up weapons and ship production were we to find ourselves in a larger conflict. In 1939 the UK was ill prepared for what it had to do, but did manage an impressive scale up of its ships and aircraft production to replace heavy losses and expand the fleets and squadrons. Training enough pilots was a bigger issue than building enough aircraft during the height of the battle of Britain.
Offering assistance to NATO requires the ability to project force away from our home base. This in turn necessitates taskforce capabilities, with air heavy lift and sea delivery to transfer personnel and weapons to the battlefield. The UK in 1914 and in 1939 on both occasions got a small professional army exposed on the continent against superior forces. The death rate in 1914 was very high and led to the need to recruit a massively larger citizens army. In 1940 the retreat from Dunkirk rescued most of our stranded army in uncomfortable surroundings with the loss of large quantities of equipment. The lesson from this is to commit in conjunction with allies in ways which improve the odds of success and reduce the likelihood of disaster from exposing too few people to too large an opposing force.
Being able to help our associated territories and countries needs that same ability to project force at a distance and to marshal sufficient force to resist an invasion or to evict an invader as we did in the Falklands. There is a similar requirement to help the UN.
As an island nation the UK will tend to have more continuous need of maritime and airpower. This can be well used in support of others when we need to intervene overseas. The UK has not tended to have a large army in peacetime, but does have a very professional and effective smaller army. We need a credible professional army for all the roles identified. This has been massively expanded during global conflicts, especially to intervene on the continent where opposing armies were large and well equipped. Now European countries are democracies and part of NATO the world has changed for the better