Precise details as to effect and things to check will depend upon the type of hot-water system installed, as the exact plumbing arrangements can vary.
With a "traditional" British system, the cold cistern in the attic supplies not only the hot-water cylinder but also the cold faucets in the bathroom, so anything nasty in the cistern would show up on those outlets too (unless the discoloration shows up only after heating).
Re bleeding the radiators, unless the heating system is extremely old, it will almost certainly be what's called an indirect type of hot-water cylinder. That means that the water which is pumped from the boiler and through the radiators doesn't actually mix with the water in the cylinder. It just goes through a coiled pipe inside the cylinder so that the heat is transferred.
If the brown is actually rust staining from the boiler/radiator circuit, then it probably means that the coil is leaking inside the cylinder. If that's the case, it won't be long before it fails completely and will need replacing.
