I drive a minibus (on a scheduled service) and needed to have a medical when I got to my 70th birthday (in July 2016).
A Doctor friend performed the examination in late April - thus leaving plenty of time.
I ended up driving (car only) during the Summer under cover of a "backstop" clause in the 1988 Road Traffic Act which allows for this.
By late August I was beginning to panic as this clause does not allow for driving abroad - and I was to do so in late September. So I contacted the DVLA and asked them to forget the Minibus entitlements (for the moment) and send me a car licence.
On the following Saturday the car licence arrived - followed by the Minibus licence on the Monday after.
This post is a very much shortened version of what happened over 4 months - missing out the numerous phone calls to Cardiff (on one occasion, when I was querying what the call-handler was telling me, she said " Oh, hang on a minute - I'll put my glasses on!!) and unnecessary trips to hospital.
But I am not alone - have a google and you will see that some people waited up to 18 months.
Maybe things have improved? The motto is (like "shop early for Xmas") - apply in plenty of time and keep pestering them.
Best of luck. Spike