There is a huge backlog at the DVLA. Many people are waiting months for their license renewal. Three weeks is relatively short. Good luck with your phone call - there are many reports of people waiting hours on repeated calls.
My understanding is, according to section 88, you can continue to drive for up to 12 months after your license expires provided you have proven approval from your doctor and that the DVLA has received your application.
Diabetes UK (the charity, NOT this forum) are in contact with DVLA about this problem. You may get assistance if you call their helpline on Tuesday.