Does one replacement automatically require another? I need the chain, chain set and cassette replaced on my road bike - does this automatically mean that the bottom bracket must also be replaced?
The short answer is no, it shouldn't. Only if it's worn out too.

With regular cleaning of the drivetrain 1 cassette usually lasts 3 chains and 1 chainring usually lasts 2-3 cassettes, assuming you replace the chain before it's too late.

Bottom brackets have no direct relation in terms of wear to the above components, but it rather depends on the following factors: quality, weather conditions, how often was it powerwashed, the terrain you ride on, your pedalling style and if it's a serviceable unit, then how often was it serviced.
A Shimano Hollowtech BB should last between 10 to 20 thousand kilometres.

Signs of a worn BB: The crankset is properly installed, but it's still loose. Clicking sound with each pedal stroke. Roughness when turning the cranks.

My personal advice: If you're worried about the BB being worn, but it doesn't have any signs, order a new one and store it in the closet until the old needs replacing. It should be an easy job to replace ( if you have the tools, otherwise take it to your LBS) and there's no need to do it together with the crankset.( Removing the crankset is 1-5 minutes, so it won't add that much extra work, but you save money and help the environment.)