How can I clear out limescale from inside a bath's pipes?
If the problem is scale sufficient to seriously reduce flow, you need a major solution and vinegar won't do.

There are acid treatments and reduced lime treatments, both very caustic chemicals that need to be handled by a pro. They are flushed through the whole system to dissolve the lime.

There is an electronic device being advertised that claims to break up the scale and put it back into solution to flush it out of the system. I have no experience with it and heard nothing about it, but it is a major installation.

The solution usually used for badly scaled pipes is replacement.  Ferrous pipes seem to build up much faster than non ferrous, but there are problems mixing metals in a system. If you plan a partial replacement, you probably want to consider plastic piping such as PEX.

Most of the solutions offered on various sites focus on reducing buildup by softening water rather than curing the problem after it occurs.  The added benefit of softened water is taste and easier rinsing of dishes and clothing.