What is correct, "in the British Islands" or "on the British Islands"? Is there a general rule?
Context, context, context! It depends on what sort of sentence you're writing; we might say "There are many countries in the British Islands" but also "On one of the British Islands there is a mountain called Pillar." We cannot possibly give a general, all-encompassing rule for this, though given a sentence we could instruct on which is correct. To decide which applies in a given context, you might consider whether the object in your sentence would most likely be described as on or in something else; a mountain rests on top of land, while a country is one of many in the group of Islands. But this is perhaps a difficult determination to make for a non-native speaker. If given a sentence as an example of your confusion, perhaps more instructive assistance could be provided.
As Stephan mentioned, you should use the word in, rather than on. Another point is that people don't normally say "British Islands", the normal term is British Isles.
The British Isles is a geographic term covering the UK, Republic of Ireland and the Isle of Man which is a British overseas territory.