290 miles approx.
Edinburgh is the westernmost city out of the three cities listed.
Edinburgh
You should check in at Birmingham at least one hourbefore your flight leaves for Edinburgh !
The driving distance between Birmingham, England and Edinburgh, Scotland is about 291 road miles.
250 Km
By aeroplane.Ryanair is the cheapest.
London, Manchester, Bath, Liverpool & Oxford
London, Manchester, Birmingham, Newcastle, York, Carlisle, Cardiff, Edinburgh, Glasgow, Dublin
246.02 miles
London, Edinburgh, Birmingham, Liverpool, York, Manchester +++
Presuming you mean Harrogate? The distance between Birmingham and Edinburgh via harrogate is 334 miles which would take approximately 6 and a half hours to drive.
There are many places where one could find information on flights from Birmingham to Edinburgh. One could try online travel sites such as Flybe for information regarding these flights.