Neither, just the UK.

British Isles refers to the whole archipelago off the north-west coast of mainland Europe, and has thousands of islands, obviously dominated by Ireland and Great Britain.

The British Empire hasn’t existed since 1997 when the UK gave Hong Kong back to China following a 99-year lease.

If you mean the UK and its overseas territories and crown dependencies, such as the Falklands and Gibraltar, then nah it doesn’t include them. They’re self-goverened and basically rely on the UK for military protection and not a huge amount more. It’s not like the French model where overseas territories are literally considered part of France.