About 4 weeks for a reply confirming my 3 year licence &asking for my old licence before they would issue my new one. The new arrived today, about another fortnight.
I'd assume any delay is between DVLA & your medical people confirming that they are happy for DLVA to issue one.
Renewed mine just after Christmas and they were surprisingly quick - about two weeks. Previous time they took about 2 months - maybe even more.
This time they didn't send the reminder or form so I rang them and got through straight away. The man put a form in the post and I received it the next day. He advised me to return it as a registered letter - they have to sign for those and usually deal with them immediately. Also had to renew my photo card which they did at the same time. Very efficient.