At handover, the dealer will normally complete the keeper change using the DVLA online system or by filling in the appropriate sections of the V5C. You should be given a green new keeper slip for your records.
The new logbook is sent directly to the address you provide, not to the dealer. It can take a couple of weeks to arrive, and sometimes longer at busy times. You can usually tax the vehicle straight away using details from the new keeper slip.
If nothing has arrived after a reasonable period, contact DVLA with your details and the vehicle registration to check the status of your V5C.
This page is neutral and factual. It is not legal advice; always check the latest official DVLA guidance.