Have you agreed with the dealer that you will get the refund for the tax disc?
With transferring the private plate you will have to get a new tax disc for the Astra with the original registration, if the dealer is sorting out the transfer then they will get both the new tax discs when they get the paperwork back from DVLA.
If they have agreed to let you apply for a refund on the Astras tax make sure they give you something in writing incase they "forget" to send you the tax disc for the Astra