Yeah, you apply for V5 from DVLA, and then you need to take it for a VIC test, £35.
Not a massive issue – but you need to be fairly confident it’s kosher – as if it’s a cut and shut, or all the numbers don’t match up you won’t get it approved. That’s without the potential of it being stolen. I’d be getting the seller to do it for you – if it’s a resto and you’re not paying much for it then maybe not.
I bought a cat C from a trader and didn’t even know about it until the DVLA wrote to me! The whole process did take quite a while though, I was driving it in the meantime but was probably 6-8 weeks to sort it.
Edit: didn’t spot it was about a bike so probably easier to tell if it’s OK or not – not sure if it’s the same process but I bet you’ll still be a similar timescale sorting it.