Only the registered keeper of a vehicle is entitled to apply to transfer its registration number.
I filled in Form V317 and took it to local DVLA Centre, together with tax discs, V5s and MoTs for both cars (ie: docs from the car it was coming off and docs for the car it was going on).
I received back new tax discs and MoTs for both cars by return of post. The V5s took longer though to come back from Swansea, roughly 5-6 weeks. The cost to transfer the plate is £80 or to put it on retention.
Just give the buyer the New Keeper Supplement V5C/2, if they need to tax it before the V5 comes back, they can use a Form V62 to it.
Also, if you need tax for either vehicle at the same time, you will need to complete a V10 tax application form or the V11 tax reminder form and provide the relevant certificate of insurance or valid cover note, and appropriate rate of vehicle tax. Please ensure that you have your vehicle tax disc details (expiry date and serial number) and vehicle identification number (VIN) or chassis number to hand.
Be aware though that the DVLA local office may wish to inspect the donor and or recipient vehicle(s).
Forms can be downloaded from
HERE
Also download Download 'United Kingdom registration certificate guidance notes' (INS160) (PDF, 202K)