If you buy a private plate from the DVLA it'll probably take a week or two to come through anyway, so I suggest as follows...
Buy your road tax before the end of this month,
Once you have bought the private plate, you don't actually receive the plates, just a V750 (Certificate of Entitlement) form, which officially entitles you to assign your reg number to a car.
You will need to take this form with your V5 & insurance certificate to your local DVLA office and they will issue you with a new tax disc with your new number on and another form (can't remember the name of it) which you take to a place where you get the plates made up!
Once you've put the plates on, just send the original tax disc back to the DVLA!
There's some helpful info here, together with links to your local DVLA office and where to get your plates made up..