As long as no criminal deception is intended, anyone can use any name they like at any time. For example, if the name on my birth certificate is John Smith, I can call myself that on Mondays but use the name Fred Bloggs on Tuesdays. On Wednesdays I could be Prince Elvis Presley the 14th and on Thursdays I could be Sharon Watson. None of those name changes will invalidate my driving licence or my insurance; it's the person who is insured, not the name. (Similarly, a woman who marries and takes her husband's surname doesn't have her licence or insurance invalidated)
However, it's obvious that having multiple names could lead to a lot of confusion, especially when the cross-checking of details is carried out on computers. It therefore makes sense that you should seek to get the error put right as soon as possible.
I suggest that you should make a photocopy of your existing licence before sending it off for correction. This is because your amended licence will have the same number. You could use the photocopy to prove that you were previously known by a different name on the licence.
It probably wouldn't be necessary to get the CBT amended. I'm not a biker but I assume that it shows the number which appears on your driving licence. (The number is probably more important than the name). However, once you've got your corrected licence, it might be sensible to call in at the test centre and explain the situation, before going for your practical test.
I'll finish by reminding you of the first words I used above: "As long as no criminal intent is intended . . ."
If you were to seek to gain a second (clean) licence, under your correct name, after getting points on a licence with your incorrect name on it, you'd be acting in contravention of those sections of the Theft Act 1968 which deal with fraud.
Chris