Done this recently.

But car was on SORN,

Went to DVLA with the New Keepers suppliment, MOT, Payment for new tax and insurance.

DVLA kept MOT and new keepers suppliment.

New MOT, Tax Disc and authorisation to get plates made certificate came through in about a week and V5 about 2 weeks later.

The MOT certificate is identical, but with the new reg and the testers signature forged by the DVLA!!!