I have reported this to BT, but I just wonder what is going on. My broadband is working fine (5Mb/s) but when I pick the phone up, there is a lot of crackles and the broadband stops. Replace the handset, and broadband works again.

What would cause the noise to only be present when the phone is off the hook?

I have tried two different phones plugged into the master socket (behind the front panel, so everything else is disconnected) and the crackles are there each time.

Try another filter.

If it still happens, then you've got a line fault

If the crackles are there when you plug into the test socket then you have a fault on the line that BT open reach are responsible for.
Make sure you contact your supplier that is responsible for the line and not your ISP.
Unless there is any damage that is not classed as fair wear and tear there should not be any charge, make sure you get the advisor that takes the fault to document that they have heard the noise.