A likely story

After all, this is the DVLC we're talking about.
You may get it back quicker by doing it on line, but waiting time is counteracted by the time (and temper) expended. It seems that you must be registered with "Government Gateway"( why?) , and you are expected to remember your details and password until you next renew your licence - in 3 years time?
Aren't your old licence and passport numbers, plus an address that hasn't changed for 40 years, enough of an ID?
Bureaucracy the only growing UK industry.