All of these side lever rook rifles were made in the Birmingham trade and retailed by many makers, H&H, Purdey and all the top names
They varied in quality and fortunately yours is one of the better ones and in excellent condition
The .360 No.5 was a successor of the heeled bullet .380 "long" or "rifle" cartridge which had a .380 bore
The two cases are almost identical but if a standard (internal) bullet was used it would have to be a .360 bore which is what most makers went for
Most people as you have, now use the .38 special cartridges I do not know of a maker of .366 bullets although Lee used to make a .365 Makarov mould but it appears very short
Good luck with your endeavours
TH44