It's a long time since I changed the tube so I went round to the electrical shop and got another tube and a starter. Got back, took the tube out, tried to put the new one in, and found that the plastic at the end was broken so that the pins are loose. Tried to put it in anyway and failed.
Tried to change the starter and found that it wouldn't come out. Because the outer casing is half melted. So I got it out eventually and tried the new starter, with the old tube, and needless to say it works perfectly well.
And it goes without saying that I reached this point ten minutes AFTER the bloody electrical shop closed.
Oh well, I'll go back and scream at them tomorrow.