Depends on the bike and if you have a sympathetic MOT station. You need a signed bill of sale, and proof that VAT was paid on the bike when it was sold new (a NOVA certificate I think) ... the latter is reasonably easy on a common model of classic bike if you get in contact with the relevant owner's register, I guess with a new-ish bike you'd have to talk to the official UK importer and you might need an EU certificate of conformity too. Also, I think copies of clear photos of the bike are required. Then you fill out the relevant form, get a V5, and buy a number plate.
I'd suggest looking up how this is done on google though, just to be on the safe side.
As for the bike itself, you can get a daytime MOT on a classic as long as you have a legal number plate (mildly undersize or flexible might be ok with some MOT stations) and an electric horn fitted - I used a battery powered one for kid's bicycles when I was in the UK. Modern bikes might need other things, but that's outside of my knowledge!