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I am not strictly a World War II historian, but the question intrigued me enough to do some research with the resources I have available.

Jack Churchill only carried his longbow during the "Phoney War," it was not something he used all the time, and from what I have read he only really used it on two occasions to cause confusion. I can't comment any more on it without speculating.

With regards to his sword, Churchill carried it in his time in the British Commandos. When British Commandos were initially formed they did not have regulations or requirements on equipment. Each Commando was supposed to supply his own rifle and pistol, but was assigned a Fairbairn–Sykes fighting knife, and sometimes a sword bayonet which was about 23 inches long. For comparison, the basket-hilted broadsword Churchill carried would only have been a bit larger than the bayonets they were issued already. They typically have a blade of about 29", about the same size as a military dress sabre. Don't imagine it as a huge Braveheart sword, it probably looked like this. In fact, you can see Jack Churchill carrying the sword in this picture. In general commandos were allowed extreme flexibility in their choice of weapons since their load-out was very mission dependent. Regulations regarding their equipment risked compromising their innate flexibility.

So his sword would only have been a bit larger than a weapon they were already issued, and it wasn't actually against a regulation to carry or use in battle given the Commando's extreme flexibility anyway. Commandos were trained to use any weapon available given that they may be forced to use enemy weapons and may often be forced to resort to hand to hand combat. Churchill could have probably made enough of an argument that the sword was technically still a part of some military dress at the time (and is today); therefore, he should be allowed to at least bring it with him to war. When suiting up for a Commando raid, I doubt many people would have told him he couldn't bring a weapon unless it risked compromising the mission. Churchill also made good use of his more typical weapons like rifle and pistol.

That said, I also can't speak to the Bagpipes, though it seems he used it mainly as a signaling device.

Sources:

Jack Churchill: Unlimited Boldness

British Commandos: 1940-46

Various Special Series publications by the US Military Intelligence Service regarding British Commandos, but primarily No. 1, from August 2, 1942. British manuals on Commandos might be more useful, but I don't have access to them.

More of the Deadliest Men Who Ever Lived for confirmation on some dates. It also had a bit about his longbow use.