Nige, if you were over Birmingham at 2000ft, especially within 'raining distance' of the Bullring, you were most certainly were in their airspace.
If you are convinced it was a drone, then surely it is your duty to airprox the incident, the airprox board's primary objective is to enhance air safety in the U.K. and they can only do that with our assistance.
You won't be the first to airprox a drone.
https://www.airproxboard.org.uk/search/?query=drone