So how did they get the black tower built?
That depends which black tower you mean.
40 Melton St (the old Railtrack House) is outside the protected view corridor; looking at St Pauls from Primrose Hill, 40 Melton St (14 stories) is well off to the right and out of the way. It is limited to this height as although it is out of the main protected view corridor, it is in the 'wider setting' corridor and must be therefore appear to be no higher than the top of St Pauls' main dome. As an aside I regard the view from the south east corner of the 13th or 14th floor as just about the best view of central London available, as there is nothing in the way to St Pauls and the city. (Guess where the CEOs office was).
1 Eversholt St is however just to the left of St Pauls, and in the corridor. It is
just clear of the main dome, however to make this tower subservient to St Pauls it is restricted to 10 stories and is lower than the bottom of the main dome. The top of the block is also level with the horizon behind, which is Blackheath. Interestingly, there is a protected view from this other direction diametrically opposite and means nothing can be built that would appear to be behind St Pauls from Blackheath.
Then there is Evergreen House (160 Euston Road). Although you can communicate from the higher floors of this building to the top floors of 1 Eversholt St by morse code (it's less than 100metres away), from Primrose Hill it is again well out of the way. It is still, however limited in height for the same reasons as 40 Melton St.
The whole subject is one that is testing the re-developers of Euston. Expect some very interesting building!